AN 


INTRODUCTION 


TO 


ALGEBRA, 


BEING 


THE  FIRST  PART 


A    COURSE    OF    MATHEMATICS, 


ADAPTED    TO    THE    METHOD    OF    INSTRUCTION    IN 
THE    AMERICAN    COLLEGES. 


BY  JEREMIAH  DAY, 

Professor  of  Mathematics  and  Natural  Philosophy,  in 
Yale  College. 


NEW-HAVEN, 

PUBLISHED    BY    HOWE    6-   DEFOREST. 


OLIVES    STEELE,    PRINTER. 

1314, 


COPV    lUCHT    SECURED. 


ADVERTISEMENT. 

"THIS  course  of  Mathematics,  when  completed? 
it  is  expected  will  consist  of  the  following  parts  : 

I.  Algebra,  now  published, 

II.  Plane  Trigonometry,  including  Analytical  Tri- 
gonometry, aad  the  nature  and  use  of  Logarithms. 

III.  The  Mensuration  of  Superficies  anH  Solids, 

IV.  Navigation  and  Surveying. 

V.  Conic  Sections. 

VI.  Spherical  Geometry  and  Trigonometry, 

VII.  Fluxions.  !••&' 

Most  of  the  succeeding  parts  will  be  considerably 
smaller  than  the  Algebra  ;  the  whole  amounting  te 
two  or  three  volumes. 


ERRATA  IJV  THE  ALGEBRA. 

JPage  5,  line  27.     For  quantity,  read  proposition. 
47,          14.     For  2o,  read  — 2a. 
101,        24.    For  -a3,  read  of  -a8, 


DISTRICT  OFWOMVECTICUT  : 

I  SEAL,  j  J-^E  it  remembered,  that  on  the  seventh  day  of 
x^_x  September,  in  the  thirty-ninth  year  of  the  Inde- 
dependence  of  the  United  States  of  America,  JEREMIAH 
DAY,  of  the  said  District,  hath  deposited  in  this  Office  the 
Title  of  a  Book,  the  right  whereof  he  claims  as  Author, 
in  the  words  following,  to  wit :  "  An  Introduction  to  Alge- 
bra, being  the  First  Part  of  a  Course  of  Mathematics, 
adapted  to  the  method  of  instruction  in  the  American 
Colleges.  By  Jeremiah  Day,  Professor  of  Mathematics 
and  Natural  Philosophy,  in  Yale  College  ;"  in  conformity 
to  the  Act  of  the  Congress  of  the  United  States,  entitled^ 
An  Act  for  the  encouragement  of  learning,  by  securing  the 
Copies  of  Maps,  Charts  and  Books,  to  the  Authors  and 
Proprietors  of  such  Copies,  during  the  times  therein  men- 
tioned, 

HENRY  W.  EDWARBS, 
Clerk  of  the  District  of 


PREFACE. 


following  summary  view  of  the  fir=t  principles  of  al- 

-  gebra,  is  intended  to  be  accommodated  to  the  method 
of  instruction  generally  adopted  in  the  AmeiieHn  colleges. 

The  books  which  have  been  published,  in  Great-Britain, 
on  mathematical  subjects,  are  principally  of  two  classes. — 
One  consists  of  extended  .treatise?,  which  enter  into  a  thor- 
ough investigation  of  the  particular  departments  which  tire 
the  objects, of  their  inquiry.  Many  of  these  are  excel- 
lent in  their  kind ;  /but  they  are  too  voluminous  for  the  use  of 
the  bedy  of  students  in  a  college. 

The  other  class  are  .expressly  intended  for  beginners;  hrt 
many  of  them  are  written  in  so  concise  a  manner,  that  im- 
portant proofs  and  illustrations  are  excluded.  They  are 
mere  fat-books,  containing  only  the  outlines  of  subjects 
which  are  to.be  explained  and  enlarged  upon,  by  the  pro- 
fessor in  his  lecture  room,  or  by  the  private  tutor  in  his 
chamber. 

In  the  colleges  in  this  country,  ihere  is  generally  put  into 
the  hands  of  a  class,  a  book  from  which  they  are  expected 
of  themselves  to  acquire  the  principles  of  the  science  to 
which  they  .are  attending ;  receiving,  however,  from  their 
instructor,  any  additional  assistance  which  may  be  found  ne- 
cessary. An  elementary  work  for  such  a  purpose,  ought  ev- 
idently to  contain  the  explanations  which  are  requisite,  to 
bring  the  subjects  treated  of  .within  the  comprehension  of 
the  body  of  the  class. 

If  the  design  of  studying^  the  mathematics  were  merely 
to  obtain  such  a  knowledge  of  the  practical  parts,  as  is  re- 
quired for  transacting  business ;  it  might  i»e  sufficient  to 
commit  to  memory  some  of  the  principal  rules,  and  to  make 
the  operations  familiar,  by  attending  to, the  examples.  In 
this  mechanical  way,  the  accountant,  the  navigator,  and  tin- 
land -surveyor,  may  be  qualified  for  their  respective  employ- 
ments, with  very  little  knowledge  of  the  principles  that  lie  at 
die  foundation  of  the  calculations  which  they  are  to  make. 

But  a  higher  object  is  proposed,  in  the  case  of  those  who 
.are  acquiring  a  liberal  education.  The  lusin  de  b.n  s 


A  %  PREFACE. 

be  to  call  into  exercise,  to  discipline,  and  to  invigorate  the 
powers  of  the  mind.  It  is  the  logic  of  the  mathematics  which 
constitutes  their  principal  value,  as  a  pail  of  a  course  of  col- 
legiate instruction.  The  time  and  attention  devoted  to  them, 
is  for  the  purpose  of  forming  sound  reasoncrs,  rather  than  ex- 
pert mathematicians.  To  accomplish  this  object,  it  is  necessary 
that  the  principles  be  clearly  explained  and  demonstrated, 
and  that  the  several  parts  be  arranged  in  such  a  manner, 'as 
to  show  the  dependence  of  one  upon  another.  The  whole 
should  be  so  conducted,  as  to  keep  the  reasoning  powers  in 
continual  exercise,  without  greatly  fatiguing  them.  No  oth- 
er subject  affords  a  better  opportunity  for  exemplifying  the 
rules  of  correct  thinking.  A  more  finished  specimen  of 
clear  and  exact  logic  has,  perhaps,  never  been  produced, 
than  the  Elements  of  Geometry  by  Euclid. 

It  may  be  thought,  by  some,  to  be  unwise  to  form  our  gen- 
eral habits  of  arguing,  on  the  model  of  a  science  in  which 
the  inquiries  are  accompanied  with  absolute  certainty  ;  while 
the  common  business  of  life  must  be  conducted  upon  proba- 
ble evidence,  and  not  upon  principles  which  admit  of  com- 
plete demonstration.  There  would  be  weight  in  this  objec- 
tion, if  the  attention  were  confined  to  the  pure  mathematics. 
But  when  these  are  connected  with  the  physical  sciences,  as- 
tronomy, chemistry,  and  natural  philosophy,  the  mind  has 
opportunity  to  exercise  its  judgment,  upon  all  the  various 
degrees  of  probability  which  occur  in  the  concerns  of  life. 

So  far  as  it  is  desirable  to  form  a  taste  for  mathematical 
studies,  it  is  important  that  the  books  by  which  the  sludent 
is  first  introduced  to  an  acquaintance  with  these  subjects, 
should  not  be  rendered  obscure  and  forbidding  by  their  con- 
ciseness. Here  is  no  opportunity  to  awaken  interest,  by  rhe- 
torical elegance,  by  .exciting  the  passions,  or  by  presenting 
images  to  the  imagination.  The  beauty  of  the  mathe- 
matics depends  on  the  distinctness  of  the  objects  of  inquiry, 
the  symmetry  of  their  relations,  the  luminous  nature  of  the 
arguments,  and  the  certainty  of  the  conclusions.  But  how  is 
this  beauty  to  be  perceived,  in  a  work  which  is  so  much 
abridged,  that  the  chain  of  reasoning  is  often  interrupted, 
many  difficulties  left  unexplained,  important  demonstrations 
omitted,  and  the  transitions  from  one  subject  to  another  so 
.abrupt,  as  to  keep  their  connections  and  dependencies  out 
of  view  ? 

It  may  not  be  necessary  to  state  every  proposition  and  its 
proof,  .with  all  the  formality  which  is  so  strictly  adhered  to 


PREFACE.  5 

.by  Euclid  ;  as  it  is  not  essential  to  a  logical  argument,  that 
it  be  expressed  in  regular  and  entire  syllogisms.  A  step  of  a 
demonstration  may  be  safely  left  out,  when  it  is  so  simple 
and  obvious,  that  no  one  possessing  a  moderate  acquaintance 
with  the  subject,  could  fail  to  supply  it  for  himself.  But  this 
liberty  of  omission  ought  not  to  be  extended,  to  cases  in 
which  it  might  occasion  obscurity  and  embarrassment.  If  it 
be  desirable  to  give  opportunity  for  the  mind  to  display  and 
enlarge  its  powers,  by  surmounting  obstacles;  full  scope 
may  be  found  for  this  kind  of  exercise,  especially  in  the 
higher  branches  of  the  mathematics,  from  difficulties  which 
will  unavoidably  occur,  without  creating  new  ones  for  the 
sake  of  perplexing. 

The  purpose  for  which  abridged  compilations  are  com- 
monly made  is,  probably,  to  save  time.  The  expense,  of  an 
additional  volume  or  two,  in  that  part  of  a  public  education 
which  is  to  occupy  a  large  portion  of  three  or  four  years, 
can  hardly  be  supposed  to  be  an  object  of  great  comparative 
importance.  The  principal  savirg  of  expense,  in  this  ca-'o, 
is  included  in  the  saving  of  time.  But  is  not  the  progress 
of  the  student  impeded,  rather  than  accelerated,  by  abridg- 
ments ?  The  time  requisite  to  become  master  of  a  subject, 
is  not  always  proportioned  to  the  number  of  pages  which  it 
occupies.  Hours  may  be  spent,  in  supplying  an  explanation, 
or  an  article  of  proof,  which,  if  it  had  been  inserted  in  its 
place,  might  haye  been  read  and  understood,  in  a  few  min- 
utes. 

Algebra  requires  to  be  treated  in  a  more  plain  and  diiTuce 
manner,  than  some  other  parts  of  the  mathematics;  because 
it  is  to  be  attended  to,  early  in  the  course,  while  the  mind  of 
the  learner  has  not  been  habituated  to  a  mode  of  thinking  so 
abstract,  as  that  which  will  now  become  necessary.  He  lias 
also  a  new  language  to  learn,  at  the  same  time  he  is  settling 
the  principles  upon  which  his  future  inquiries  are  to  be  con- 
ducted. These  principles  ought  to  be  established,  in  the 
most  clear  and  satisfactory  manner  which  the  nature  of  the 
case  will  admit  of.  Algebra  and  geometry  may  be  consider- 
ed as  lying  at  the  foundation  of  the  succeeding  branches  of 
the  mathematics,  both  pure  and  mixed.  Euclid  and  others 
have  given  to  the  geometrical  part,  a  degree  of  clearness 
and.  precision  which  would  be  very  desirable,  but  is  hardly 
to  be  expected,  in  algebra. 

For  the  reasons  which  have  been  mentioned,  the  manner 
in  which  the  following  pages  are  written,  is  not  the  iuo-:t 


«  PREFACE. 

concise.  But  the  work  is  necessarily  limited  in  extent 
of  subject.  It  is  far  from  being  a  complete  treatise  of  alge- 
bra. It  is  merely  an  introduction.  It  is  intended  to  con- 
lain  as  much  matter,  as  the  student  at  college  can  nttend  to, 
with  advantage,  during  the  short  time  allotted  to  this  partic- 
ular study.  There  is  generally  but  a  small  portion  of  a  class, 
who  have  either  leisure  or  inclination,  to  pursue  mathematic- 
al inquiries  much  farther,  than  is  necessary  to  maintain  an 
honourable  standing,  in  the  institution  of  which  they  are  mem- 
bers. Those  few  who  have  an  unusual  taste  for  this  science, 
and  aim  to  become  adepts  in  it,  ought  to  be  referred  to  sepa- 
rate and  complete  treatises,  on  the  different  branches.  No 
one  who  wishes  to  be  thoroughly  versed  in  mathematics, 
should  look  to  compendiums  and  elementary  books,  for  any 
thing  more,  than  the  first  principles.  As  soon  as  these  arc 
acquired,  he  should  be  guided  in  his  inquiries,  by  the  genius 
and  spirit  of  original  authors. 

In  the  selection  of  materials,  those  articles  have  been  ta- 
ken which  have  a  practical  application,  which  are  not  of 
very  difficult  comprehension,  and  wiiich  are  preparatory  to 
succeeding  parts  of  the  mathematics,  philosophy,  and  astron- 
omy. The  object  has  not  been,  to  introduce  original  matter. 
In  the  mathematics,  which  have  been  cultivated  with  success, 
from  the  days  of  Pythagoras,  and  in  which  the  principles  al- 
ready established  are  sufficient  to  occupy  the  most  active 
mind  for  years,  the  parts  to  which  the  student  ought  firm  to 
attend,  are  not  those  recently  discovered.  Free  use  has 
been  made  of  the  works  of  Newton,  Maclaurin,  Saunderson, 
Simpson,  Euler,  Emerson,  Lacroix,  and  others,  but  in  a  way 
that  rendered  it  inconvenient  to  refer  to  them,  in  particular 
instances.  The  proper  field  for  the  display  of  mathematical 
genius,  is  in  the  region  of  invention.  But  what  is  requisite 
for  an  elementary  work,  is  to  collect,  arrange,  and  illustrate, 
materials  already  provided.  However  humble  this  employ- 
ment, he  ought  patiently  to  submit  to  it,  whose  object  is  to 
instruct,  not  those  who  have  made  considerable  progress  in 
the  mathematics,  but  those  who  are  just  comraencino-  the 
study.  Original  discoveries  are  not  for  the  benefit  of  oegin- 
yiers,  though  they  may  be  of  great  importance  to  the  ad- 
vancement of  science. 

The  arrangement  of  the  parts  is' such,  that  the  explanation 
of  one  is  not  made  to  depend  on  another  which  is  to  follow. 
The  addition,  multiplication,  and  division  of  powers,  for  in- 
stance, is  placed  after  involution.  In  the  statement  of  gemv 


trd\  rules,  if  they  are  reduced  to  a  small  number,  their  appli- 
cations to  particular  cases  may  not,  always,  be  readily  under- 
stood. On  the  other  hand,  if  they  are  very  numerous,  they 
become  tedious  and  burdensome  to  the  memory.  The  rules 
given,  in  tkis  introduction,  are  most  of  them  comprehensive; 
but  they  are  explained  and  applied,  in  subordinate  articles. 

A  particular  demonstration  is  sometimes  substituted  for  a 
general  one,  when  the  application  of  the  principle  to  other 
cases  is  obvious.  The  examples  are  not  often  taken  from 
philosophical  subjects,  as  the  learner  is  supposed  to  be  famil- 
iar with  none  of  the  sciences  except  arithmetic.  In  treat- 
ing of  negative  quantities,  frequent  references  are  made  to 
mercantile  concerns,  to  debt  and  credit,  &c.  These  are  mere- 
ly for  the  purpose  of  illustration.  The  whole  doctrine  of 
negatives  is  made  to  depend  orr  the  single  principle,  that 
they  are  quantities  to  be  subtracted.  But  the  student,  at 
this  early  period,  is  not  accustomed  to  abstraction.  He  re- 
quires particular  examples,  to  catch  his  attention,  and  aid  his 
conceptions. 

The  section  on  proportion  will,  perhaps,  be  thought  use- 
less to  those  who  read  the  fifth  book  of  Euclid.  That  is 
sufficient  for  the  purposes  of  pure  geometrical  demonstration. 
Bat  it  is  important  that  the  propositions  should  also  be  pre- 
sented, under  the  algebraic  forms.  In  addition  to  this,  great 
assistance  may  be  derived  from  the  algebraic  notation,  fn  de- 
monstrating, and  reducing  to  system,  the  laws  of  proportion. 
The  subject,  instead  of  being  broken  up  into  a  multitude  of 
distinct  propositions,  may  be  comprehended  in  a  few  gene- 
ral principles. 


CONTENTS. 

Page. 
Introductory  Observations,  on  the  Mathematics  in 

general,  1 

ALGEBRA. 

Section  I.  Notation,  Positive  and  Negative  quali- 
ties, Axioms,  &c.  10 

II.  Addition^  28 

III.  Subtraction,  35 

IV.  Multiplication,  39 

V.  Division,  51 

VI.  Algebraic  Fractions,  60 

VII.  Reduction  of  Equations,  by  Transposi- 

tion, Multiplication,  and  Division.  So- 
lution of  Problems,  78 

VIII.  Involution.     Notation,   addition,  sub- 

traction^ multiplication,  and   division, 

of  Powers,  94 

IX.  Evolution.  Notation,  redaction,  addition, 

subtraction,    multiplication,    division, 

and  involution,  of  Radical  Quantities,       111 

X.  Reduction  of  Equations  by  Involntion  and 

Evolution.  Affected  Quadratic  Equations,      1 3(> 

XI.  Solution  of  Problems  which  contain  two 

or  mere  Unknown  Quantities,  1 5& 

XII.  Ratio  and  Proportion,  175 

XIII.  Variation,  or  General  Proportion,  204 

XIV.  Arithmetical  and  Geometrical  progression,  -1 1 
XV.-  Mathematical  Infinity,  '  220 
XV  f.  Division  by  Compound  Divisors,  227 

XVII.  Involution   of  compound   quantities, 

by  the  Binomial  Theorem,  232 

XVIII.  Evolution  of  Compound  Quantities,       24^ 

XIX.  Infinite  Series,  246 

XX.  Composition  and  Resolution  of  the  high- 

er Equations,  254 

XXI.  Application  of  Algebra  to  Geometry,         262 

XXII.  Equations  of  Curves  275. 


INTRODUCTORY  OBSERVATIONS 

•%*.• 


ON     THB 


MATHEMATICS  IN  GENERAL, 


ART.  1.    TtfATHEMATIjCS  ii>Ae  science  of  QUAN- 

TITY. 

Any  thing  which  can  be  increased  or  diminished,  or 
which  is  capable  of  being  measured,  is  called  quantity. 
Thus,  a  line  is  a  quantity,  because  it  can  be  made  long- 
er or  shorter;  and  can  be  measured,  by  applying  to  it 
another  line,  as  a  foot,  a  yard,  or  an  ell.  Weight  is  a, 
quantity,  which  can  be  measured,  in  pounds,  ounces, 
and  grains.  Time  is  a  species  of  quantity,  whose  mea- 
sure can  be  expressed,  in  hours,  minutes,  and  seconds. 
But  colour  is  not  a  quantity.  It  cannot  be  said,  with 
propriety,  that  one  colour  is  either  greater  or  less  than 
another.  The  operations  of  the  mind,  such  as  thought, 
choice,  desire,  hatred,  &c.  are  not  quantities.  They 
are  incapable  of  mensuration. 

2.  Those  parts  of  the  Mathematics,  on  which  all 
the  others  are  founded,  are  Arithmetic,  Algebra,  and 
Geometry. 

3.  ARITHMETIC  is  the  science  of  numbers.     Its  aid 
is  required,  to  complete  and  apply  the  calculations,  in 
almost  every  other  department  of  the  mathematics. 

B 


2  MATHEMATICS. 

4.  ALGEBRA  is  a  method  of  computing  principally 
by  letters.      FLUXIONS  may  be  considered  as  belong- 
ing lo  the  higher  branches  of  algebra. 

5.  GEOMETRY  is  that  part  of  the  mathematics,  which 
treats  of  magnitude.     By  magnitude, in  the  appropri- 
ate sense  of  the  term,  is  meant  that  species  of  quan- 
tity, which  is  extended]  that  is,  which  has  one  or  more 
of  the  three  dimensions,  length,  breadth,  and  thickness. 
Thus,  a  line  is  a  magnitude,  because  it  is  extended,  in 
length.     A  surface  is  a  magnitude,  having  length  and 
breadth.  Asolid  is  a  magnitude,  having  length,  breadth, 
and.  thickness.    But  motion,  though  a  quantity,  is  not, 
strictly  speaking,  a  magnitude.     It  has  neither  length, 
breadth,  nor  thickness.* 

6.  TRIGONOMETRY  and  CONIC  SECTIONS  are  branch- 
es of  the  mathematics,  in  which,  the  principles  of  ge- 
ometry are  applied  to  triangles,  and  the  sections  of  a 
cone. 

7.  Mathematics  are  either  pure,  or  mixed.     Inpure 
mathematics,  quantities  are  considered,  independent- 
ly of  any  substances  actually  existing.     But,  in  mixed 
mathematics,  the  relations  of  quantities  are  investiga- 
ted, in  connection   with  some  of   the  properties  of 
matter,  or  with  reference  to  the  common  transactions 
of  business.    Thus,  in  Surveying,  mathematical  prin- 
ciples are  applied  to  the  measuring  of  land;  in  Op- 
tics, to  the  properties  of  light;  and  in  Astronomy,  to 
the  motions  of  the  heavenly  bodies. 

8.  The  science  of  the  pure  mathematics  has  long 
been  distinguished,  for  the  clearness  and  distinctness 
of  its  principles ;  and  the  irresistible  conviction,  which 
they  carry  to  the  mind  of  every  one  who  is  once  made 
acquainted  with  them.     This  is  to  be  ascribed,  partly 
to  the  nature  of  the  subjects,  and  partly  to  the  ex- 
actness of  the  definitions,  the  axioms,  and  the  demon- 
strations. 

*  NOTE.     Some  writers,  however,  use  magnitude,  as  sy- 
nonymous with  quantity. 


MATHEMATICS.  3 

$.  The  foundation  of  all  mathematical  knowledge 
must  be  laid,  in  definitions  and  selfevident  truths.  A 
definition  is  an  explanation  of  what  is  meant,  by  any 
word  or  phrase.  Thus,  an  equilateral  triangle  is  de- 
fined, by  saying,  that  it  is  a  figure  bounded  by  three 
equal  sides. 

It  is  essential  to  a  complete  definition,  that  it  per- 
fectly distinguish  the  thing  defined,  from  every  thing 
else.  On  many  subjects,  it  is  difficult  to  give  such 
precision  to  language,  that  it  shall  convey,  to  every 
hearer  or  reader,  exactly  the  same  ideas.  But,  in  the 
mathematics,  the  principal  terms  may  be  so  defined^ 
as  not  to  leave  room  for  the  least  difference  of  appre- 
hension, respecting  their  meaning.  All  must -be 
agreed,  as  to  the  nature  of  a  circle,  a  square,  and  a 
triangle,  when  they  have  once  learned  the  definitions 
of  these  figures. 

Under  the  head  of  definitions,  may  be  included  ex- 
planations of  the  characters  which  arc  used  to  denote 
the  relations  of  quantities.  Thus,  the  character  -y/  is 
explained  or  defined,  by  saying  that  it  signifies  the 
same  as  the  words  square  root. 

10.  The  next  step,  after  becoming  acquainted  with 
the  meaning  of  mathematical  terms,  is  to  bring  them 
together,  in  the  form  of  propositions.     Some  of  the 
relations  of  quantities  require  no   process  of  reason- 
ing, to  render  them  evident.     To  be  understood,  they 
need  only  to  be  proposed.     That  a  square  is  a  differ- 
ent figure  from  a  circle ;  that  the  whole  of  a  thing  is 
greater,  than  one  of  its  parts;  and,  that  two  straight 
lines  cannot  inclose  a  space,  are  propositions  so  mani- 
festly true,  that  no  reasoning  upon  them  could  make  . 
them  more  certain.     They  are,  therefore,  called  self- 
evident  truths,  or  axioms. 

11.  There  are,  however,  comparatively  few  mathe- 
matical truths  which  are  selfevident.     Most  require 
to  be  proved,  by  a  chain  of  reasoning.     Propositions 
of  this  nature  are  denominated  theorems;    and   the 
process,  by  which,  they  are  shewn  to  bo  true.,  is  called 


4  MATHEMATICS. 

demonstration.  This  is  a  mode  of  arguing,  in  which, 
.ev«ry  inference  is  immediately  derived,  either  from 
definitions  and  selfevidcut  axioms,  or  from  principles 
which  have  been  previously  demonstrated.  In  this 
way,  complete  certainty  is  made  to  accompany  every 
step,  in  a  long  course  of  reasoning. 

12.  Demonstration  is  either  direct,  or  indirect.  The 
former  is  the  common,  obvious  mode  of  .conducting 
a  demonstrative  argument     But,  in  some  instances, 
it  is  necessary  to  resort  to  indirect  demonstration; 
which  is  a  method  of  establishing  a  proposition,  by 
.proving  that  to  suppose  it  not  true,  would  lead  to  an 
absurdity.     This  is  frequently  called  reductio  ad  absur- 
dum.     Thus,  in  certain  cases  in  geometry,  two  lines 
may  be  proved  to  be  equal,  by  shewing  that  to  sup- 
pose them  unequal,  would  involve  an  absurdity. 

13.  Besides  the  principal  theorems  in  the  mathe- 
matics, there  are   also  Lemmas,  and  Corollaries.     A 
Lemma  is  a  proposition  which  is  demonstrated,  for 
•the  purpose  of  using  it,  in  the  demonstration  of  some 
other  proposition.    This  preparatory  step  is  taken,  to 
prevent  the  proof  of  the  principal  theorem  from  be- 
roming  complicated  and  tedious. 

14.  A  Corollary  is  an  inference  from  a  preceding 
proposition.     A  Scholium  is  a  remark  of  any  kind, 
suggested  by  something    which    has    gone   before, 
though  not,  like  a  corollary,  immediately  depending 
on  it. 

15.  The  immediate  object  of  inquiry,  in  the  math- 
ematics, is,  frequently,  not  the  demonstration  of  a 
general  truth,  but  a  method  of  performing  some  op- 
eration, .such  as  reducing  a  vulgar  fraction  to  a  decim- 
al, extractiiig  the  cube  root,  or  inscribing  a  circle  in 
a  square.     This  is  called  solving  a  problem.     A  theo- 
rem is  something  to  be  proved.     A  problem  is  some- 
thing to  be  done. 

16.  When  that  which  is  required  to  be  done,  is  so 
easy,  as  tt>  be  obvious  to  every  one,  without  an  ex- 
£>lajiation,  it  is  called  a  postulate.    Of  this  nature,  is 


MATHEMATICS.  5 

ihe  drawing  of  a  straight  lino,  from  one  point  to  an- 
other. A  postulate  is,  to  a  problem,  what  an  axiom 
is,  to  a  theorem. 

17.  A  quantity  is  said  to  be  given,  when  it  is  either 
supposed  to  be  already  known,  or  is  made  a  condition, 
in  the  statement  of  any  theorem  or  problem.     In  the 
rule  of  proportion  in  arithmetic,  for  instance,  three 
terms  must  be  given,  to  enable  us  to  find  a  fourth. — 
These  three  terms  are  the  data,  upon  which  the  cal- 
culation is  founded.     If  we  are  required  to  find  the 
number  of  acres,  in  a  circular  island  ten  miles  in  cir- 
cumference, the  circular  figure,  and  the  length  of  the 
circumference,  are  the  data.     They  are  said  to  be 
given  by  supposition,  that  is,  by  the  conditions  of  the 
problem.     A  quantity  is  also  said  to  be  given,  when  it 
maybe  directly  and  easily,  inferred,  from  something 
else  wThich  is  given.     Thus,  if  two  numbers  are  giv- 
en, their  sum  is  given;   because  it  is  obtained,  by 
^merely  adding  the  numbers  together. 

In  Geometry,  a  quantity  may  be  given,  either  in 
fosition,  or  magnitude,  or  both.  A  line  is  given  in 
position,  when  its  situation  and  direction  are  known. — 
It  is  given  in  magnitude,  when  its  length  is  known.  A 
circle  is  given  in  position,  when  the  place  of  its  centre 
is  known.  Jt  is  given  in  magnitude,  when  the  length 
of  its  diameter  is  known. 

18.  One  quantity  is  contrary,  or  contradictory  to 
another,  when,  what  is  affirmed,  in  the  one,  is  denied, 
in  the  other.     A  proposition  and  its  contrary,  can 
never  both  be  true.     It  cannot  be  true,  that  two  given 
lines  are  equal,  and  that  they  are  not  equal,  at  the 
same  time. 

19.  One  proposition  is   the   converse   of  another, 
when  the  order  is  inverted ;  so  that,  what  is  given  or 
supposed,  in   the  first,   becomes   the   conclusion,   in 
the  last ;  and  what  is  given  in  th.e  last,  is  the  conclu- 
sion, in  the  first.     Thus,  it  can  be  proved,  first,  that  if 
the  sides  of  a  triangle  are  equal,  the  angles  are  equal; 
and  secondly,  that  if  the  angles  are  equal,  the  sides 


e  MATHEMATICS 

are  equal.  Here,  in  the  first  proposition,  the  equal- 
ity of  the  sides  is  given  ;  and  the  equality  of  the  angles, 
inferred:  in  the  second,  the  equality  of  the  angles  is 
given,  and  the  equality  of  the  sides  inferred.  In  ma- 
ny instances,  a  proposition  and  its  converse  are  both 
true;  as  in  the  preceding  example.  But  this  is  not 
always  the  case.  A  circle  is  a  figure  bounded  by  a 
curve;  but  a  figure  bounded  by  a  curve  is  not  of 
course  a  circle. 

20.  The  practical  applications  of  the  mathematics, 
in  the  common  concerns  of  business,  in  the  useful 
arts,  and  in  the  various  branches  of  physical  science, 
are  almost  innumerable.  Mathematical  principles  are 
necessary,  in  mercantile  transactions,  for  keeping,  ar- 
ranging, and  settling  accounts,  adjusting  the  prices  of 
commodities,  and  calculating  the  profits  of  trade  :  in 
Navigation,  for  directing  the  course  of  a  ship  on  the 
ocean,  adapting  the  position  of  her  sails  to  the  direc- 
tion of  the  wind,  finding  her  latitude  and  longitude, 
and  determining  the  bearings  and  distances  of  objects 
on  shore  :  in  Surveying,  for  measuring,  dividing,  and 
laying  out  grounds,  taking  the  elevation  of  hiUs,  and 
fixing  the  boundaries  of  fields,  estates  and  public  ter- 
ritories :  in  Mechanics,  for  understanding  the  laws  of 
motion,  the  composition  of  forces,  the  equilibrium  of 
the  mechanical  powers,  and  the  structure  of  machines: 
•in  Arcliiiecture,  for  calculating  the  comparative  strength 
of  timbers,  the  pressure  which  each  will  be  required  to 
sustain,  the  forms  of  arches,  the  proportions  of  .col- 
umns, &c. :  in  Fortification,  for  adjusting  the  position, 
lines,  and  angles,  of  the  several  parts  of  the  wqrks : 
in  Gunnery,  for  regulating  the  elevation  of  the  can- 
non, the  force  of  the  p'owder,  and  the  velocity  and 
range  of  the  shot :  in  Optics,  for  tracing  the  direction 
of  the  rays  of  light,  understanding  the  formation  of 
images,  the  laws  of  vision,  the  separation  of  colours, 
the  nature  of  the  rainbow,  and  the  construction  of  mi- 
croscopes and  telescopes  :  in  Astronomy,  for  compu- 
ting the  distances,  magnitudes,  and  revolutions  of  the 


MATHEMATICS.  7 

heavenly  bodies;  and  the  influence  of  the  law  of  grav- 
itation, in  raising  the  tides,  disturbing  the  motions  of 
the  moon,  causing  the  return  of  the  comets,  and  re- 
taining the  planets  in  their  orbits :  in  Geography, 
for  determining  the  figure  and  dimensions  of  the 
earth,  the  extent  of  oceans,  islands,  continents,  and 
countries  ;  the  latitude  and  longitude  of  places,  th,e 
courses  of  rivers,  the  height  of  mountains,  and  the 
boundaries  of  kingdoms  :  in  History,  for  fixing  the 
chronology  of  remarkable  events,  and  estimating  the 
strength  of  armies,  the  wealth  of  nations,  the  value  of 
their  revenues,  and  the  amount  of  their  population  : 
and,  in  the  concerns  of  Government,  for  apportioning 
taxes,  arranging  schemes  of  finance,  and  regulating 
national  expenses.  The  mathematics  have  also  im- 
portant applications  to  Chemistry,  Mineralogy,  Music, 
Painting,  Sculpture,  and  indeed  to  a  great  proportion 
of  the  whole  circle  of  arts  and  sciences. 

21.  It  is  true,  that,  in  many  of  the  branches  which, 
have  been  mentioned,  the   ordinary  business  is  fre- 
quently  transacted,   and  the   mechanical  operations 
performed,  by  persons  who  have  not  been  regularly 
instructed  in  a  course  of  mathematics.     Machines  are 
framed,  lands  are  surveyed,  and  ships  are  steered,  by 
men  who  have  never  thoroughly  investigated  the  prin- 
ciples, which  lie  at  the  foundation  of  their  respective 
arts.     The  reason  of  this  is,  that  the  methods  of  pro- 
ceeding, in  their  several  occupations,  have  been  point- 
ed out  to  them,  by  the  genius  and  labour   of  others. 
The   mechanic  often  works  by  rules,  which  men  of 
science'  have  provided,  for  his  use,  and  of  which  he 
knows  nothing  more,  than  the  practical  application. 
The  mariner  calculates  his  longitude  by  tables,  for 
which  he  is  indebted  to  mathematicians  and  astrono- 
mers  of  no  ordinary  attainments.      In  this  manner, 
even  the  abstruse  parts  of  the  mathematics  are  made 
to  contribute  their  aid,  to  the  common  arts  of  life. 

22.  But  an  additional  and  more  important  advan- 
tage, to  persons  of  a  liberal  education,  is  to  be  found, 


8  MATHEMATICS. 

in  the  enlargement  and  improvement  of  the  reason- 
ing powers.  The  mind,  like  the  body,  acquires 
strength  by  exertion.  The  art  of  reasoning,  like  other 
arts,  is  learned  by  practice.  It  is  perfected,  only  by 
long  continued  exercise.  Mathematical  studies  are 
peculiarly  fitted  for  this  discipline  of  the  mind.  They 
are  calculated  to  form  it  to  habits  of  fixed  attention  ; 
of  sagacity,  in  detecting  sophistry  ;  of  caution,  in  the 
admission  of  proof ;  ofdexterity,  in  the  arrangement 
of  arguments  ;  and  of  skill,  in  making  all  the  parts  of 
a  long  continued  process  tend  to  a  result,  in  which  the 
truth  is  clearly  and  firmly  established.  When  a  habit 
of  close  and  accurate  thinking  is  thus  acquired ;  it  may 
be  applied  to  any  subject,  on  which  a  man  of  letter* 
or  of  business  may  be  called  to  employ  his  talents. 
"  The  youth,"  says  Plato,  "  who  are  furnished  with 
mathematical  knowledge,  are  prompt  and  quick,  at  all 
other  sciences." 

It  is  not  pretended,  that  an  attention  to  other  ob- 
jects of  inquiry,  is  rendered  unnecessary,  by  the  study 
of  the  mathematics.  It  is  not  their  office,  to  lay  be- 
fore us  historical  facts ;  to  teach  the  principles^  of 
morals  ;  to  store  the  fancy  with  brilliant  images  ;  or 
to  enable  us  to  speak  and  write  with  rhetorical  vigour 
and  elegance.  The  beneficial  effects  which  they  pro- 
duce on  the  mind,  are  to  be  seen,  principally,  in  the 
regulation  and  increased  energy  of  the  reasoning  pow- 
trs.  These  they  are  calculated  to  call  into  frequent 
and  vigorous  exercise.  At  the  same  time,  mathe- 
matical studies  may  be  so  conducted,  as  not  often  .to 
require  excessive  exertion  and  fatigue.  Beginning 
wim  the  more  simple  subjects,  and  ascending  gradu- 
ally to  those  which  are  more  complicated  ;  the  mind 
acquires  strength,  as  it  advances  ;  and  by  a  succession 
of  steps,  rising  regularly  one  above  another,  is  ena- 
bled to  surmount  the  obstacles  which  lie  in  its  way. 
In  a  course  of  mathematics,  the  parts  succeed  each 
other  in  such  a  connected  series,  that  the  preceding 
propositions  are  preparatory  to  those  which  follow. 


MATHEMATICS.  9 

The  student  who  has  made  himself  master  of  the 
former,  is  qualified  for  a  successful  investigation  of 
the  latter.  But  he  who  has  passed  over  any  of  the 
ground  superficially,  will  find  that  the  obstructions  to 
his  future  progress  are  yet  to  be  removed.  In  math- 
ematics, as  in  war,  it  should  be  made  a  principle,  not 
to  advance,  while  any  thing  is  left  unconquered  be- 
hind. It  is  important  that  the  student  should  be 
deeply  impressed  with  a  conviction  of  the  necessity 
of  this.  Neither  is  it  sufficient  that  he  understands 
the  nature  of  one  proposition  or  method  of  opera- 
tion, before  proceeding  to  another.  He  ought  also 
to  make  himself  familiar  with  every  step,  by  a  careful 
attention  to  the  examples.  He  must  not  expect  to 
'become  thoroughly  versed  in  the  science,  by  merely 
reading  the  main  principles,  rules  and  observations. 
It  is  practice  only,  which  can  put  these  completely  in 
his  possession.  The  method  of  studying,  here  recom- 
mended, is  not  only  that  which  promises  success,  but 
that  which  will  be  found,  in  the  end,  to  be  the  most 
expeditious,  and  by  far  the  most  pleasant.  While  a 
superficial  attention  occasions  perplexity  and  conse- 
quent aversion  ;  a  thorough  investigation  is  rewarded 
with  a  high  degree  of  gratification.  The  peculiar  en- 
tertainment which  mathematical  studies  are  calculated 
to  furnish  to  the  mind,  is  reserved  for  those  who  make 
themselves  masters  of  the  subjects  to  which  their  at- 
tention is  called. 


NOTE.  The  principal  definitions,  theorems,  rules,  fee. 
which  it  is  necessary  to  comrhit  to  memory,  are  distinguished 
by  being  put  in  Italics. 


ALGEBRA, 

SECTION  I. 

Notation,  Negative  Quantities,  Axioms,  fye. 

A  2T  ALGEBRA  may  be  defined,  a  general 
"^  method  of  investigating  the  relations  of 
quantities,  principally  by  letters.  This,  it  must  be  ac- 
knowledged, is  an  imperfect  account  of  the  subject;, 
us  every  account  must  necessarily  be,  which  is  com- 
prised in  the  compass  of  a  definition.  Its  real  nature 
>s  to  be  learned,  rather  by  an  attentive  examination 
of  its  parts,  than  from  any  summary  description. 

The  solutions  in  Algebra,  are  of  a  more  general 
nature,  than  those  in  common  Arithmetic.  Ihe  lat- 
ter relate  to  particular  numbers  ;  the  former,  to  whole 
classes  of  quantities.  On  this  account,  Algebra  has 
been  termed  a  kind  of  universal  Arithmetic.  The 
generality  of  its  solutions  is  principally  owing  to  the 
use  of  letters,  instead  of  numeral  figures,  to  express 
the  several  quantities  which  are  subjected  to  calcula- 
tion. In  Arithmetic,  when  a  problem  is  solved,  the 
answer  is  limited  to  the  particular  numbers  which  are 
specified,  in  the  statement  of  the  question.  But  an 
algebraic  solution  may  be  equally  applicable  to  all 
other  quantities  which  have  the  same  relations.  This 
important  advantage  is  owing  to  the  difference  be- 
tween the  customary  use  of  figures,  and  the  manner 
in  which  letters  are  employed  in  Algebra.  One  of 
the  nine  digits  invariably  expresses  the  same  number : 
Hut  a  letter  may  be  put  for  any  number  whatever. 
The  figure  8  always  signifies  eight ;  the  figure  5,  five, 
&c.  And,  though  one  of  the  digits,  in  connection  with 


ALGEBRA.  II 

wlhcrs,  may  have  a  local  value,  different  from  its  sim- 
ple value  when  alone  ;  yet  the  same  combination  al- 
ways expresses  the  same  number.  Thus  263  has  one 
uniform  signification.  And  this  is  the  case  with  eve- 
ry other  combination  of  figures.  But  in  Algebra,  a 
letter  may  stand  for  any  quantity  which  we  wish  it  to 
represent.  Thus  b  may  be  put  for  &  or  10,  or  50,  or 
1 000.  It  must  not  be  understood  from  this,  however, 
that  the  letter  has  no  determinate  value-.  Its  value  is 
fixed  for  the  occasion.  For  the  present  purpose,  it 
remains  unaltered.  But  on  a  different  occasion,  the 
same  letter  may. be  put  for  any  other  number. 

A  calculation  may  be  greatly  abridged  by  the  use 
of  letters ;  especially  when  very  large  numbers  are 
concerned.  And  when  several  such  numbers  are  to 
be  combined,  as  in  multiplication,  the  process  becomes 
extremely  tedious.  But  a  single  letter  may  be  put 
fora  large  number,  as  well  as  for  a  small  one.  Ihe 
numbers  26347297,  68347823,  and  27462498,  for  in- 
stance, may  be  expressed  by  the  letters  b,  c  and  d. 
The  multiplying  them  together,  as  wiH  be  seen  here- 
after, will  be  nothing  more,  than  writing  them,  one  af- 
ter another,  in  the  form  of  a  word,  and  the  product 
will  be  simply  led.  Thus,  in  Algebrannluch  of  the  la- 
bour of  calculation  may  be  saved,  by  the  rapidity  of 
the  operations.  Solutions  are  sometimes  effected,  in 
the  compass  of  a  few  lines,  which,  in  common  Arith- 
metic, must  be  extended  through  many  pages. 

24.  Another  advantage  obtained  from  the  notation 
by  letters  instead  of  figures,  is,  that  the  several  quan- 
tities which  are  brought  into  a  calculation,  may  be 
^preserved  distinct  from  each  other,  though  carried 
through  a  number  of  complicated  processes ;  where- 
as, in  arithmetic,  they  are  so  blended  together,  that 
no  trace  is  left  of  what  they  were,  before  the  opera- 
tion began.  To  give  a  very  simple  example :  sup- 
pose it  is  required  to  multiply  together  74,  23  and 
41.  By  arithmetic,  the  product  is  found  to  be  69782. 
This  product,  however,  would  not  of  itself  suggest 


12  ALGEBRA. 

the  numbers  which  had  been  multiplied  together  t» 
produce  it.  But  in  algebra,  if  74  be  represented  by 
a,  23,  by  6,  and  41,  by  c;  the  product  will  be  ab<-, 
which  not  only  stands  for  69782;  but  also  indicates 
the  factors,  a,  6,  and  c,  which  were  multiplied,  to  ob- 
tain this  product. 

25.  Algebra  differs  farther  from  arithmetic,  in 
making  use  of  unknown  quantities,  in  carrying  on  its 
operations.  In  arithmetic,  all  the  quantities  which 
enter  into  a  calculation  must  be  known.  For  they 
are  expressed  in  numbers.  And  every  number  must 
necessarily  be  a  determinate  quantity.  But  in  alge- 
bra, a  letter  may  be  put  for  a  quantity,  before  its 
value  has  been  ascertained.  And  yet  it  may  have 
such  relations,  to  other  quantities,  with  which  it  is 
connected,  as  to  answer  an  important  purpose  in  the 
calculation. 


NOTATION. 

26.  To  facilitate  the  investigations  in  algebra,  the 
several  steps  of  the  reasoning,  instead  of  being  ex- 
pressed in  word^  are  translated  into  the  language  of 
signs  and  symbols,  which  may  be  considered  as  a 
species  of  short-hand.   This  serves  to  place  the  quan- 
tities and  their  relations  distinctly  before  the  eye,  and 
to  bring  them  all  into  view  at  once.     They  are  thus 
more  readily  compared  and  understood,  than  when 
removed  at  a  distance  from  each  other,  as  in  the 
common  mode  of  writing.     But  before  any  one  can 
avail  himself  of  this  advantage,  he  must  become  per- 
fectly familiar  with  the  new  language. 

27.  The  quantities  in  algebra,  as  has  been  already 
observed,  are  generally  expressed  by  letters.     The 

first  letters  of  the  alphabet  are  used,  to  represent 
known  quantities;  and  the  last  letters,  those  which  are 
unknown.  Thus  b  may  be  put  for  a  known,  and  y,  for 
an  unkno'wn  quantity.  Sometimes  the  quantities,  in- 


NOTATION.  13 

stead  of  being  expressed  by  letters,  are  set  down  in 
figures,  as  in  common  arithmetic. 

28.  Besides  the  letters,  and  figures,  there  are  cer- 
tain characters  used,  to  indicate  the  relations  of  the 
quantities,  or  the   operations   which   are   performed 
with  them.     Among  these  are  the  signs  -j-  and  — , 
which  are  read  phis  and  minus,  or  more  and  less.   The 
former  is  prefixed  to  quantities  which  are  to  be  added; 
the  latter,  to  those  which  are  to  be  subtracted.     Thus 
a  -f-  b  signifies  that  b  is  to  be  added  to  a.    If  a  stands 
for   10,  and  b,  for  6;  then  a  4-  b  is  16.     It  is  read  a 
plus  6,  or  a  added  to  b,  or  a  and  6.     If  the  expression 
be  a  —  6,  i.  e.  a  minus  b;   it  indicates  that  b  is  to  be 
subtracted  from. a.     In  figures  10  —  6  is  10  diminish- 
ed by  G  i.  e.  4. 

29.  The  sign  +  is  prefixed  to  quantities  which  arc 
considered  as  affirmative  or  positive;  and  the  sign  — , 
to  those  which  are  supposed  to  be  negative.    For  the 
nature  of  this  distinction;  see  art.  54. 

All  the  quantities  which  enter  into  an  algebraic 
process,  are  considered,  for  the  purposes  of  calcula- 
tion, as  either  positive  or  negative.  Before  the  first 
one,  unless  it  be  negative,  the  sign  is  generally  omit- 
ted. But  it  is  always  to  be  understood.  Thus  a  +  b^ 
is  the  same  .as  -f  «  +  b.  For  a,  is  as  much  added 
to  by  as  b  is,  to  a.  They  are  added  together:  a  +  b 
is  a  and  b;  or,  which  is  the  same  thing,  b  and  a. 

30.  Sometimes  both  +  and  —  are  prefixed  to  the 
same  letter.     The  sign  is  then  said  to  be  ambiguous. 

Thus  a  _  b  signifies  that  in  certain  cases,  compre- 
hended in  a  general  solution,  b  is  to  be  added  to  c, 
and,  in  other  cases,  subtracted  from  it. 

31.  When  it  is  intended  to  express  the  difference 
between  two  quantities  without  deciding  which  is  the 
one  to  be  subtracted,  .the  character  cf>  or  ^  is  used. 
Thus  a  -f  6,  or  «  tj)  b  denotes  the  difference  between 
«  and  b,  without  determining  whether  a  is  to  be  sub- 
tracted from  6,  or  b  from  a. 


14  .  ALGEBRA, 

32.  The  equality  between  two  quantities  or  sets  of 
quantities  is  expressed,  by  parallel  lines  =.      Thus 
a  +  6  =  d  signifies  that  a  and  6  togetliMp  are  equal 
to  d.     So  8  +  3  =  11  i.  e.  8  and  3  equal  11.     And 
a  +  d  =  C  =  b+g  =  h  signifies  that  a  and  d  equal 
e,  which  is  equal  to  6  and  g,  which  are  equal  to  h. — 
Again  8  +  4=  1C  -4=  10  +  2  =  7  +  2  +  3 
=  12.     This  is  read  8  +  4  is  equal  to  16  —  4,  which 
is  equal  to  10  +  2,  which  is  equal  to  7  +  2  +  3,. 
which  is  equal  to  12. 

33.  When  the  first  of  the  two  quantities  compared 
is  greater,  than  the  other,  the  character  >  is  placed 
between  them.     Thus  a  >  b  signifies  that  a  is  great- 
er than  b. 

If  the  first  is  less  than  the  other,  the  character  < 
is  used;  as  a  <  b;  i.  e.  a  is  less  than  b.  In  both  ca- 
ses, the  quantity  towards  which  the  character  opens, 
is  greater  than  the  other. 

34.  A  numeral  figure  is  often  prefixed  to  a  letter. 
This  is  called  a  co-efficient.     It  shows  how  often  the 
quantity  expressed  by  the  letter  is  to  be  taken.   Thus 
26  signifies  twice  6,  and  96,  9  times  6,  or  9  multiplied 
into  6.     If  6  stands  for  10,  then  96  is  9  times  10  or 
90. 

The  co-efficient  may  be  either  a  whole  number  or 
a  fraction.  Thus  f  6  is  two  thirds  of  6.  When  the 
co-efficient  is  not  expressed,  1  is  always  to  be  under- 
stood. Thus  a  is  the  same  as  1  a  i.  e.  once  a. 

35.  The  co-efficient  may  be  a  letter,  as  well  as  a 
figure.     In  the  quantity  m6,  m  may  be  considered  the 
co-efficient  of  6;  because  6  is  to  be  taken  as  many 
times  as  there  are  units  in  m.     If_5  stands  for  6,  then 
mb  is  6  times  6.     In  3o6c,  3  may  be  considered  as 
the  co-efficient  of  abc;  3 a,  the  co-efficient  of  be; 
or  3  a  6,  the  co-efficient  of  c.     See  art.  42. 

36.  A  simple  quantity  is  either  a  single  letter  or 
number,  or  several  letters  connected  together,  with- 
out the  signs  +  and   — .      Thus  a,  06,  abd,  and  86 
are  each  of  them   simple   quantities.     A  compound 


NOTATION.  15 

quantity  consists  of  a  number  of  simple  quantities, 
connected  Iw  the  sign  +  or  — „  Thus  a  +  6,  d  —  y, 
b  —  d  -f-  3  A,  are  each  compound  quantities.  The 
members,  of  which  it  is  composed,  are  called  terms. 

37.  If  there  are  two  terms  in  a  compound  quan- 
tity, it  is  called  a  binomial.    Thus  «  +  &  and  a  —  b  are 
binomials.     The  latter  is  also  called  a  residual  quan- 
tity, because  it  expresses  the  difference  of  two  quan- 
tities, or  the  remainder,  after  one  is  taken  from  the 
other.      A  compound    quantity   consisting   of  three 
terms,  is  sometimes  called  a  trinomial ;  one  of  four 
terms,  a  quadrinomial,  &-C. 

38.  When  the  several  members  of  a  compound 
quantity  are  to  be  subjected  to  the  same  operation, 
they  are  frequently  connected  by  a  line  called  a  vincu- 
lum.    Thus  a  —  b  +  c  shows  that  the  sum  of  b  and 
c  is  to  be  subtracted  from  a.     But  a  —  b  -f  c  signifies 
that  b  only  is  to  be  subtracted  from  er,  while  c  is  to  be 
added.     The  sum  of  c  and  rf,  subtracted  from  the 

sum  of  a  and  i,  is  a  +  b  —  c  +  d.  The  marks  used 
for  parentheses,  (  )  are  often  substituted,  instead  of 
a  line,  for  a  vinculum.  Thus  x  —  (a  -\-  c)  is  the 
same  as  x  —  a  +  c.  The  equality  of  two  sets  of 
quantities  is  expressed,  without  using  a  vinculum. — 
Thus  a  +  b  =  c  +  d  signifies,  not  that  b  is  equal  to 
c;  but  that  the  sum  of  a  and  b  is  equal  to  the  sum  of 
c  and  d. 

39.  A*single  letter,  or  a  number  of  letters,  repre- 
senting any  quantities  with  their  relations,  is  called  an 
algebraic  expression;  and  sometimes  a.  formula.  Thus 
«  +  b  +  3</  is  an  algebraic  expression. 

\^  40.  The  character  x  denotes  multiplication.  Thus 
a  x  b  is  a  multiplied  into  b :  and  6  x  3  is  6  times  3, 
or  6  into  3.  Sometimes  a  point  is  used  to  indicate 
multiplication.  Thus  a  .  b  is  the  same  as  a  x  b.  But 
the  sign  of  multiplication  is  more  commonly  omitted, 
between  simple  quantities;  and  the  letters  are  con-* 
wected  together,  in  the  form  of  a  word  or  syllable. — 


16  ALGEBRA 

Thus  ab  is  the  same  as  a .  b  or  a  x  b.  And  bcde  is 
the  same  as  6  x  c  x  d  x  e.  When  a  compound  quan- 
tity is  to  be  multiplied,  a  vinculum  is  usea,  as  in  the 
case  of  subtraction.  Thus  the  sum  of  a  and  6,  mul- 
tiplied into  the  sum  of  c  and  d,  is  a  -f  b  x  c  +  rf,  or 
(a  +  6)  x  (c  +  <*).  And  (6  +  2)  x  5  is  8  x  5  or 
40.^  ButG  +  2  x  5  is  6  +  10  or  16.  When  the 
marks  of  parenthesis  are  used,  the  sign  of  multiplica- 
tion is  frequently  omitted.  Thus  (a?  +  y)  (#  —  y)  is 
(x  +  y)  x  (x  —  y). 

41.  When  two  or  more  quantities  are  multiplied 
together,  each  of  them  is  called  a  factor.     In  the 
product  a  6,  a  is  a  factor,  and  so  is  b.    In  the  product 

x  x  «  +  m,  x  is  one  of  the  factors,  and  a  +  m,  the 
other.  Hence  every  co-efficient  may  be  considered  a 
factor.  (Art.  35.)  In  the  product  3  y,  3  is  a  factor, 
j)S  well  as  y. 

42.  A  quantity  is  sfcid  to  be  resolved  into  factors, 
when  any  factors  are  taken  which,-  being  multiplied 
together,  will  produce  the  given  quantity.     Thus  3  a  b 
may  be  resolved  into  the  two  factors  3  a  and  b  be- 
cause 3  a  x  6  is  3  a  6.      And  5  a  win  may  be  resolved 
into  the  three  factors  5  a,  and  m,  andn;  because  5  a 
X  m  x  n  is  5amn.      And  48  may  be  resolved,  into 
the  two  factors  2  x  24,   or  3  x  16,  or  4  x  12,  or 
C  X  8;  or  into  the  three  factors  2  x  3  X  8,  or  4  x  G 
X  2,  &c. 

43.  The   character  ~-   is  used  to  show,  that  the 
quantity  which  precedes  it,  is  to  be  divided,  by  that 
which   follows.      Thus  a  ~-  c  is  a  divided  'by  c:  and 

a  +  b  -7-  c  -|-  d  is  the  sum  of  a  and  i,  divided  by 
the  sum  of  c  and  rf.  But  in  algebra,  division  is  morfc 
commonly  expressed,  by  writing  the  divisor  under  the 
dividend,  hi  the  form  of  a  vulgar  fraction.  Thus 
a  c  *r  b 

7  is  the  same  as  a  4-  b :  and  v~i~7  is  the  difference  of 
b  d+h 

c  and  b  divided  by  the  sum  of  d  and  A.  A  charac- 
ter prefixed  to  the  dividing  line  of  a  fractional  ex- 


NOTATION,  17 

prcssion,  b  lo  be  understood  as  referring  to  all  the 
parts  taken  collectively;  that  is,  to  the  whole  value  of 

b  4-  c 

the  quotient.     Thm  a  —  — ; —    siorufies    that    the 

m  +  n      ° 

quotient  of  b  +  c  divided  by  m  +  n  is  to  be  subtract- 

c  —  d       k  +  n 

ed  from  o.     And  — T~~  X  — * denotes  that  the 

a+  m       x  —  y 

first  quotient  is  to  be  multiplied  into  the  second. 

44.  When    four   quantities   are  proportional  y  the 
proportion  is  expressed  by  points,  in  the  same  man- 
ner, as  in  the  Rule  of  Three  in  arithmetic.      Thus 
«  :  b  : :  c  :  d  signifies  that  a  has  to  b,  the  same  ratio, 
which  c  has  to  d.     And*  a  6  :  c  d  : :  a  -f*  m  :  b  +  w, 
means,  that  a  b  is  to  c  d ;  as  the  sum  of  a  and  m,  to 
the  sum  of  b  and  n. 

45.  Algebraic  quantities  are  said  to  be  alike,  when 
they  are  expressed  by  the  same  letters,  and  are  of  the 
szmepoiecr:  and  unlike,  \vhe»  the  letters  are  differ- 
ent, or  when  the  same  letter  is   raised   to   different 
powers.*  Thus  ab,3a  b,  —  a  b,  and  —  Ga  b,  are  like 
quantities,  because  the  letters  are  the  same  in  each, 
although  the  signs  and  co-efficients  are  different.   But 
So,  3y,  and  36  x,  are  unlike   quantities,  because  the 
letters  are  unlike,  although  there  is  no  difference  in 
the  signs  and  co-efficients. 

46.  One  quantity  is  said  to  be  a  multiple  of  an- 
ethcr,  when  the  former  contains  the  latter  a  certain 
number  of  times,  without  a  remainder.      Thus  10  a 
is  a  multiple  of  2 a,  because   the   one   contains  the 
ether  just  5  times.     So  2-fls  a  multiple  of  G. 

47.  One  quantity  is  said  to  be   a  measure   of  an- 
other, when  the  former  is  contained  in  the  latter,  any 
Dumber  of  times,  without  a  remainder.      Thus  36  is 
a.-measjB-e  of  156:  and  7  is  a  measure  of  35. 

48.  Whe  value,  of  an  expression,  is  the  number  or 


*For  the  notation  of  powers  and  raolst  see  the,  sections. 
those  subjects. 

D 


IS  ALGEBRA. 

quantity,  for  which  the  expression  stands.      Thus  the 
value  of  3  +  4  is  7;  of  3  x  4  is  12;  of  V  is  2. 

49.  The  RECIPROCAL  of  a  quantity^  is  the  quotient 
arising  from  dividing  an  unit  by  that  quantity.      Thus 

the  reciprocal  of  a  is  —  ;  the  reciprocal  of  «  -f  I  n 

—  qr-r  ;  the  reciprocal  of  4  is  —£ 

50.  The  relations  of  quantities,  which,  in  ordinary 
language,  are  signified  by  words,  are  represented,  in 
ihe  algebraic  notation,  by  signs.     The  latter  mode  of 
expressing  these  relations,  ought  to  be  made  so  fa- 
miliar to  the  mathematical  student,  that  he  can,  at 
any  time,  substitute  the  one  for  the  other.      A  few 
examples  are  here  added,  in  which,  words  are  ta  be 
converted  into  signs. 

1.  What  is  the  algebraic  expression  for  the  fol- 
lowing statement,  in  which,  the  .  letters  a,  b,  c,  &c. 
may  be  supposed  to  represent  any  given  quantities  ? 

The  product  of  a,  b,  and  c,  divided  by  the  differ- 
ence of  c  and  d,  is  equal  to  the  sum  of  frand  c  added 
to  15  times  h. 
ale 

Ans. 


2.  The  tfroduct  of  the  difference  of  a  and  7t  into 
the  sum  01  6,  c,  and  d,  is  equal  to  37  times  m,  added 
to  the  quotient  of  b  divided  by  the  sum  of  A  and  b.  — 
Ans. 

3.  The   sum  of  a  and  b,  is  to  the  quotient  of  b  di- 
vided by  c;   as  the  product  of  a  into  c,  to  12  times  h. 
Ans. 

4.  The  sum  of  a,  b,  and  c  divided  by   six   times 
their  product,  is  equal  to  four  times  their  sum  dimin- 
ished by  d.     Ans. 

5.  The  quotient  of  6  divided  by  the  sum  of  «  and 
7>,  is  equal  to  7  times  d,  diminished  by  the  quotient  of 
b,  divided  by  36.   ,  Ans. 

51.  It  is  necessary  also,  to  be  able  to  reverse  what 


NOTATION.  19 

is  done  in  the  preceding  examples,  that  is,  to  trans- 
late the  algebraic  signs  into  common  language. 

What  will  the  fallowing  expressions  become,  when 
words  are  substituted  for  the  signs  ? 
a  -\-  b  a 

1  .  —  r  —  —  abc  —  6  w  +  —  ;  —  . 
n  a  -\-  c 

Ans.  The  sum  of  a  and  b  divided  by  h,  is  equal  to 
the  product  of  «,  b,  and  c,  diminished  by  6  times  mt 
and  increased  by  the  quotient  of  a  divided  by  the 
sum  of  a  and  c. 

3h-c  _  h 


Ans. 
•  c  _  (jd 

3.  a+l(h  +  x}  -2^-+4=(a  +  A)x(ft-<0. 

;     Ans. 

d  —  4  . 

4.  a  —  6  :  a  c : : :  3  X  h  +  d  +  y. 

m 

Ans. 


'3  +  6  wi  am  +dm 

Ans. 

• 

52.  At  the  close  of  an  algebraic  process,  it  is  fre- 
quently necessary  to  restore  the  numbers,  for  which 
letters  had  been  substituted,  at  the  beginning.  In  do- 
ing this,  the  sign  of  multiplication  must  not  be  omit- 
ted, as  it  generally  is,  between  factors  expressed  by 
letters.  Thus  if  a  stands  for  3,  and  b,  for  four;  the 
product  o  b  is  not  34,  but  3  x  4  i.  c.  12. 


20  ALGEBRA. 

In  the  foil  owing  examples, 

Let  a  —  3  And  d  —  6. 

5=4  tn  —  8. 

c  =  2  n  =  10. 

<z  +  m,        ic  — n      3+8      4x2—10 
Then,  1.  --j-  +  -gj-  =  ^--g  +  Tx"8~~~' 


a  6  —  3 rf        36n  --  be 
3.bmd  +  —d—    -  4tf  +  3c7 


53.  An  algebraic  expression,  in  which  numbers 
have,  in  this  manner,  been  substituted  for  letters,  may 
often  be  rendered  much  more  simple,  by  reducing 
several'  terms  to  one.  This  can  not  generally  be 
done,  while  the  letters  remain.  If  a  +  6  is  used  fur 
the  sum  of  two  quantities,  a  can  not  be  united  in  tho 
same  term  with  b.  But  if  a  stands  for  3,  and  6,  for  4, 
then  a  +  6  =  3  +  4==7.  The  value  of  an  express 
sion  consisting  of  many  terms  may  thus  be  found,  by 
actually  performing,  with  the  numbers,  the  operations 
of  addition,  subtraction,  multiplication,  &c.  indicated 
%y  the  algebraic  characters. 

Find  the  value  of  the  following  expressions,  in 
which,  the  letters  are  supposed  to  stand  for  the  same 
numbers,  as  in  the  preceding  article. 

ad  ^  X  6 

1.    —  +  a  +  m  n  =  — ^     +  3  +  8  x  10  =  0 

+  3  +  80  =  92. 
2J 


.  « 

m  —  u 

2  x  10  =  120. 


3.  a+cxn-—  m  +     ~H~    —  a  X  n  —  m  =  6. 


NEGATIVES.  21 


4    L2Ll±L£  +  a6c-l±* 

n  —  a 

ac  +  5m 
5*     2n  +  3"" 


POSITIVE  AND  NEGATIVE  QUANTITIES.* 

54.  To  one  who  has  just  entered  on  the  study  of 
•algebra,  there  is  generally  nothing  more  perplexing, 
than  the  use  of  what  are  called  negative  quantities. — 
He  supposes  he  is  about  to  be  introduced,  to  a  class 
of  quantities,  which  are  entirely  new;  a  sort  of  math- 
ematical nothings,  of  which  he  can  form  no  distinct 
conception.     As  positive  quantities  are  real,  he  con- 
cludes that  those  which  are  negative  must  be  imogtw- 
ary.    But  this  is  owing  to  a  misapprehension  of  the 
term  negative,  as  used  in  the  mathematics. 

55.  A  NEGATIVE  quantity  is  one  which  is  required  to 
be  SUBTRACTED.     When  several  quantities  enter  into 
a  calculation,  it  is  frequently  necessary  that  some  of 
them  should  be  added  together,  while  others  are  sub- 
tracted.     The  former  are  called  affirmative  or  posi- 
tive, and  are  marked  with  the  sign  •$- ;  the  latter  are 
termed  negative,  and  distinguished  by  the  sign  — . 
If,  for  instance,  the  profits  of  trade  are  the  subject  of 
calculation,  and  the  gain  is  considered  positive ;  the 
loss  will  be  negative ;  because  the  latter  must  be  sub- 
tracted from  the  former,  to  determine  the  clear  profit. 
If  the  sums  of  a  book  account,  are  brought  into  an 
algebraic  process,  the  debt  and  the  credit  are  distin- 
guished by  opposite  signs.     If  a  man  on  a  journey  is, 
by  any  accident  necessitated  to  return  several  miles, 

*  On  the  subject  of  Negative  quantities,  see  Newton's  Uni- 
versal Arithmetic,  Maseres  on  the  Negative  Sign,  Mansfield's* 
Mathematical  Essays,1!  and  Maclaurin'?.  Simpson's,  Iviilcr's,- 
Sattnderson's  and  Ludkm's  Algebra. 


*2  ALGEBRA. 

tliia  backward  motion  is  to  be  considered  negative, 
because  that,  in  determining  his  real  progress,  it  must 
be  subtracted,  from  the  distance  which  he  has  travel- 
led in  the  opposite  direction.  If  the  ascent  of  a  body 
from  the  earth  be  called  positive,  its  descent  will  be 
negative.  These  are  only  different  examples  of  the 
same  general  principle.  In  each  of  the  instances, 
one  01  the  quantities  is  to  be  subtracted  from  the 
other. 

56.  The  terms  positive  and  negative,  as  used  in  the 
mathematics,  are  merely  relative.      They  imply  tlwt 
there  is,  either  in  the  nature  of  the  quantities,  or  iu 
their  circumstances,  or  in  the  purposes  which  they 
are  to  answer  in  calculation,  some  such  opposition  as 
requires  that  one  should  be  subtracted  from  the  other. 
But  this  opposition  is  not  that  of  existence  aid  non- 
existence,  nor  of  one  thing  greater  than  nothing,  and 
another  less  than  nothing.      For,  in  many  cases,  ci- 
ther of  the  signs  may  be,  indifferently  and  at  pleas- 
ure, applied  to  the  very  same  quantity;  that  is,  the 
IAO  characters  may  change  places.     In  determining 
the  progress  of  a  ship,  for  instance,  her  easting  may 
be  marked  -{-,  and  ner  westing  — ;  or  the  westing 
may  be  +,  and  the  easting  — .     All  that  is  necessary 
is,  that  the  two  signs  be  prefixed  to  the  quantities,  in 
such  a  manner  as  to  shovv,  which  are  to  be  addeo^, 
and  which  subtracted.      In  different  processes,  they 
may  be   differently   applied.      On  one   occasion,  a 
downward  motion  may  be  called  positive,  and  on  an- 
ethcr  occasion,  negative. 

57.  In  every  algebraic  calculation,  some  one  of  the 
quantities  must  be  fixed  upon,  to  be  considered  posi- 
tive.     All  other  quantities  which  will  increase  this, 
»iust  be  positive  also.      But  those  which  will  tend  to 
diminish  it,  must  be  negative.     In  a  mercantile  con- 
cern, if  the  stock  is  supposed  to  be  positive,  the  profits 
will  be  positive;  for  tney  increase  the  stock;  they  are 
to  be  added  to  it.     But  the  losses  will  be  negative ; 
for  they  diminish  the  stock;  they  are  to  be  subtracted 


NEGATIVES.  26 

from  it.  When  a  boat,  in  attempting  t%.  ascend  a 
river,  is  occasionally  driven  back  by  the  current;  if 
the  progress  up  the  stream,  to  any  particular  point,  is 
considered  positive,  every  succeeding  instance  of  for- 
ward motion  will  be  positive,  while  the  backward  mo- 
tion will  be  negative. 

58.  A  negative  quantity  is  frequently  greater,  than 
the  positive  one  with  which  it  is  connected.      But 
how.  it  may  be  asked,  can  the  former  be  subtracted 
from  the  latter  ?   The  greater  is  certainly  not  contain- 
ed in  the  less :  how  then  can  it  be  taken  out  of  it  ? 
The  answer  to  this  is,  that  the  greater  may  be  suppo- 
sed first  to  exhaust  the  less,  and  then  to  leave  a  re- 
mainder equal  to  the  difference  between  the  two.     It 
a  man  has  in  his  possession,  1000  dollars,  and  ha* 
contracted  a  debt  of  1500;  the  latter  subtracted  from 
the  former,  not  only  exhausts  the  whole  of  it,  but 
leaves  a  balance  of  .500  against  lum.       In  common 
language,  he  is  500  dollars  worse  than  nothing. 

59.  In  this  way,  it  frequently  happens,  in  the  courre 
of  an  algebraic  process,  that  a  negative  quantity  is 
brought  to  stand  alone.   It  has  the  sign  of  subtraction, 
without  being   connected   with  any  other   quantity, 
from  which  it  is  to  be  subtracted.     This  denotes  that, 
a  previous  subtraction  has  left  a  remainder,  which  is 
a  part  of  the  quantity  subtracted.      If  the  latitude  of 
a  ship  which  is  20  degrees  north  of  the  equator,  is 
considered  positive,  and  if  she  sails  south  25  degrees ; 
her  motion  first  diminishes  her  latitude,  then  reduces 
it  to  nothing,  and  finally  gives  her  5  degrees  of  south 
latitude.      The  sign  — ,  prefixed  to  tlie  25  degrees  is 
retained  before  the  5,  to  show  th.it  this  is  what  re- 
mains of  the  southward  motion,  after  balancing  the 
20  degrees  of  north  latitude.      If  the  motion  south- 
ward is  only  15  degrees,  the  remainder  must  be  -f-  5, 
instead  of  —  5,  to  show  that  it  is  a  part  of-  the  ship's 
northern  latitude,  which  has  been  thus  far  diminished, 
but  not  reduced  to  nothing.     The  balance  of  a  book 
account  will  be  positive  or  negative,  according  as  the 


U  ALGEBRA. 

debt  or  the  credit  is  the  greater  of  the  two.  To  de- 
termine to  which  side  the  remainder  belongs,  the  sign 
jnust  be  retained,  though  there  is  no  other  quantity, 
from  which  this  is  again  to  be  subtracted,  or  to  which 
it  is  to  be  added. 

GO.  When  a  quantity  continually  decreasing  is  re- 
duced to  nothing,  it  is  sometimes  said  to  become  af- 
terwards less  than  nothing.  But  this  is  an  exception- 
able manner  of  speaking.*  No  quantity  can  be  real- 
ly less  than  nothing.  It  may  be  diminished,  till  it 
vanishes,  and  gives  place  to  an  opposite  quantity.  The 
latitude  of  a  ship  crossing  the  equator,  is  first  made 
less,  then  nothing,  and  afterwards  contrary  to  what  it 
was  before.  The  north  and  south  latitudes  majf 
therefore  be  properly  distinguished,  by  the  signs  -j- 
and  —  ;  all  thtf  positive  degrees  being  on  one  side  of 
0,  and  all  the  negative,  on  the  other;  thus, 
•f  6,  -f  5,  +4,  +  3,  +  1,0,  -  1,  -2,  -  3,  - 4,-5,fcc. 

The  numbers  belonging  to  any  other  series  of  op^- 
posite  quantities,  may  be  arranged  in  a  similar  man- 
ner. So  that  0  may  be  conceived  to  be  a  kind  of 
dividing  point  between  positive  and  negative  numbers* 
On  a  thermometer,  the  degrees  above  0  may  be  con- 
sidered positive,  and  those  below  0,  negative. 

Gl.  A  quantity  is  sometimes  said  to  be  subtracted 
from  0.  By  this  is  meant,  that  it  belongs  on  the  nega- 
tive side  of  0.  But  a  quantity  is  said  to  be  added  to  0, 
when  it  belongs  on  the  positive  side.  Thus  in  speak- 
ing of  the  degrees  of  a  thermometer,  0  -f  6  means  6 
degrees  above  0;  and  0  —  6,  6  degrees  below  0. 

*  Nora.  The  expression  "less  ilian  nothing"  may  not  be 
wholly  improper;  if  it  i«  intended  to  be  understood,  not  lite- 
rally, but  merely  as  a  convenient  phrase  adopted  for  the  sake 
of  avoiding  a  tedious  circumlocution  ;  as  we  say  "tlie  sun  ri- 
ses," instead  of  aayirrg  ';  the  <>arlh  rolls  round,  and  brings  tltf: 
sun  into  vievr."  The  use  of  it  inlbi*  manner,  is  warranted  by 
Newton,  Euler,  antf  others. 


ALGEBRA.  25 


AXIOMS. 

C2.  The  object  of  mathematical  inquiry  is,  generally,  to 
investigate  some  unknown  quantity,  and  discover  how  great 
it  is.  *  This  is  effected,  by  .comparing  it  with  some  other 
quantity  or  quantities  already  known.  The  dimensions  of 
a  stick  of  timber  are  found,  by  applying  to  it  a  measuring 
rule  of  known  length.  The  weight  of  a  body  is  ascertained, 
by  placing  it  in  one  scale  of  a  balance,  and  observing  how 
many  pounds  in  the  opposite  scale,  will  equal  it.  And  any 
quantity  is  determined,  when  it  is  found  to  be  equal  to 
some  known  quantity  or  quantities. 

Let  a  and  b  be  known  quantities,  and  y,  one  which  is  un- 
known. Then  y  will  become  known,  if  it  is  discovered  to 
be  equal  to  the  sum  of  a  and  b :  that  is,  if 

y  -=r  a  -f  b, 

An  expression  like  this,  representing  the  equality  between 
one  quantity  or  set  of  quantities,  and  another,  is  called  an 
equation.  It  will  be  seen  hereafter,  that  much  of  the  busi- 
ness of  algebra  consists  in  finding  equations,  in  which,  some 
unknown  quantity  is  shown  to  be  equal  to  others  which  are 
known.  But  it  is  not  often  the  fact,  that  the  first  compari- 
son of  the  quantities,  furnishes  the  equation  required.  It 
will  generally  be  necessary  to  make  a  number  of  additions, 
subtractions,  multiplications,  &c.  before  the  unknown  quan- 
tity is  discovered.  But  in  all  these  chaages,  a  constant 
equality  must  be  preserved,  between  the  two  sets  of  quanti- 
ties compared.  This  will  be  done,  if  in  making  the  altera- 
tions, we  are  guided  by  the  following  axioms.  These  are 
not  inserted  here,  for  the  purpose  of  being  proved;  for 
they  are  self-evident.  (Art.  10.)  But  as  they  must  be  con- 
tinually introduced  or  implied,  in  demonstrations  and  the  so- 
lutions of  problems,  they  are  placed  together,  for  the  con- 
venience of  reference. 

63.  Axiom  1.  If  the  same  quantity  or  equal  quantities 
be  added  to  equal  quantities,  their  sums  will  be  equal. 

2.  If  the  same  quantity  or  equal  quantities  be  subtracted 
from  equal  quantities,  the  remainders  will  be  equal. 

3.  If  equal   quantities  be  multiplied  into  tjie  same,  or 
equal  quantities,  the  products  will  be  equal. 

4.    If  equal  quantities  be  divided  by  the  same  or  equal 
quantities,  the  quotients  will  be  equal. 
E 


UO  ALGEWiA. 

•5.  If  the  same  quantity  be  both  added  to  and  subtracted 
from  another,  the  value  of  the  latter  will  not  be  altered. 

0.  If  a  quantity  be  both  multiplied  and  divided  by  another, 
the  value  of  the  former  will  not  be  altered. 

7.  If  to  unequal  quantities,  equals  be  added,  tire  greater 
will  give  the  greater  sum. 

8.  If  from  unequal  quantities,  equals  be  subtracted,  the 
greater  will  give  the  greater  remainder. 

y .  1  f  itaequal  quantities  be  multipled  by  equals,  the  great- 
er will  give  the  greater  product. 

10.  If  unequal  quantities  be  divided  by  equals,  the  great- 
er will  give  the  greater  quotient. 

1 1 .  (Quantities  which  are  respectively  equal  to  any  other 
quantity,  are  equal  to  each  other. 

1 2.  The  whole  of  a  quantity  is  greater  than  a  part. 

This  is,  by  no  means  a  complete  list  of  the  self-evident 
propositions,  which  are  furnished  by  the  mathematics.  It  is 
not  necessary  to  enumerate  them  all.  Those  have  been  se- 
lected, to  which  we  shall  have  the  most  frequent  occasion  to 
refer. 

If  a=lo  Andiffl>ce? 

T^ien  by  ax.l.  a  +  m=-lc-{-tn  By  ax.T.  a+m>  cd+m 

ax.2.  a—m—bc—m  ax.8.  a— m>  cd—m 

az.3.  um=bcm  ax.9.  am>  cdm 

a      be  a     cd 

fl.v.4.— =— 
m      m 

ax.S.a  —a+m—tn 
am 

l=~m 


64.  The  investigations  in  algebra  are  carried  on  principal- 
ly, by  means  of  a  series  of  equations  and  proportions.  But 
instead  of  entering  directly  upon  these,  it  will  be  necessary 
to  attend,  in  the  first  place,  to  a  number  of  processes,  on 
which  the.  management  of  equations  and  proportions  de- 
pends. These  preparatory  operations  are  similar  to  the  cal- 
culations under  the  common  rules  of  arithmetic.  We  have 
addition,  multiplication,  division,  involution,  &c.  in  algebra, 
as  welj,  as  in  arithmetic.  But  this  application  of  a  common 
wame,  to  operations  in  these  two  branches  of  the  mathcmat- 


ALGEBRA.  g: 

ics,  is  often  the  occasion  of  perplexity  and  mistake.  The 
learner  naturally  expects  to  find  addition  in  algebra  the  same 
as  addition  in  arithmetic.  They  are  in  fact  the  same,  in 
many  respects :  in  all  respects  perhaps,  in  which  the  steps  of 
the  one  will  admit  of  a  direct  comparison,  with  those  of  the 
other.  But  addition  in  algebra  is  more  extensive,  than  in 
arithmetic.  The  same  observation  may  bo  made,  concern- 
ing several  other  operations  in  algebra.  They  are,  in  many 
points  of  view,  the  same  as  those  which  bear  the  same  names 
in  arithmetic.  But  they  are  frequently  extended  farther, 
and  comprehend  processes  which  are  unknown  to  arithme- 
tic. This  is  commonly  owing  to  the  introduction  of  nega- 
tive quantities.  The  management  of  these  requires  steps 
which  are  unnecessary,  where  quantities  of  one  class  only 
are  concerned.  It  will  be  important  therefore,  as  we  pass 
along,  to  mark  the  difference,  r.s  well  as  the  resemblance,  be- 
tween arithmetic  and  algebra;  and,  in  some  instances,  to 
give  a  new  definition,  accommodated  to  the  latter. 


• 

SECTION  II. 

ADDITION. 

-    • 

*        pr    TN  entering  on  an  algebraic  calculation,  the  first 

thing  to  be  done,  is  evidently  to  collect  the  ma- 
terials. Several  distinct  quantities  are  to  be  concerned  in 
the  process.  These  must  be  brought  together.  They  must 
be  connected  in  some  form  of  expression,  which  will  pre- 
sent them  at  once  to  our  view,  and  show  the  relations  which 
they  have  to  each  other.  This  collecting  of  quantities  is 
what,  in  algebra,  is  called  addition.  It  may  be  defined,  the 
connecting  of  several  quantities,  with  their  signs,  in  one  alge- 
braic expression. 

66.  It  is  common  to  include  in  the  definition,  "uniting  in 
one  term,  such  quantities,  as  will  admit  of  being  united/1 
But  tliis  is  not  so  much  a  part  of  the  addition  itself,  as  a 
reduction,  which  accompanies  or  follows  it.  The  addition 
may,  in  all  cases,  be  performed,  by  merely  connecting  the 
quantities,  by  their  proper  signs.  Thus  a  added  to  b  is,  evi- 
dently, a  and  6  :  that  is,  according  to  the  algebraic  notation 
a  +  b.  And  a,  added  to  the  sum  of  b  and  c,  is  a-f-  b+c. 


And  a+6,  added  tojc+e?,  is  a+b  +  c-\-d.  In  the  same  man- 
ner, if  the  sum  of  any  quantities  whatever,  be  added  to  the 
sum  of  any  others,  the  expression  for  the  whole,  will  contain 
all  these  quantities,  connected  by  the  sign  +  . 

Thus  the  sum  of  a-j-26  and  4c-f-  d-\-h  and  w+y,  is 


67.  Again,  if  the  difference  of  a  and  b  be  added  to  c;  the 
sum  will  be  a  —  b  added  to  c,  that  is  a—  b  +  c.  And  if  a  —  b 
be  ^dded  te  c  —  d,  the  sum  will  be  a—  b  +  c—  d.  In  one  of 
the  compound  quantities  added  here,  a  is  to  be  diminished 
by  6,  and  in  the  other,  c  is  to  be  diminished  by  d;  the  sum 
of  a  and  c  must  therefore  be  diminished,  both  by  b}  and  by 
d,  that  is,  the  expression  for  the  the  sum  total,  must  contain 
—  b  and  —  d.  On  the  same  principle,  all  the  quantities 
»'hich,  in  the  parts  to  be  added,  have  the  negative  sign,  must 


ADDITION.  29 

retain  this  sign,  in  the  amount.      Thus  a+2b—c,  added  to 
d—  h  —  m,is  a  +  2b  —  c+d—-h—m. 

68.  The  sign  must  be  retained  also,  when  a  positive  quan- 
tity is  to  be  added,  to  a  single  negative  quantity.  If  a  be  added 
to  —  b,  the  sinn  will  be  —  b  +  a.  Here  it  may  be  objected, 
that  the  negative  sign  prefixed  to  b,  shows  that  it  is  to  be 
subtracted.  What  propriety  then  can  there  be  in  adding  it  ? 
In  reply  to  this,  it  may  be  observed,  that  the  sign  prefixed 
to  b  while  standing  alone,  signifies  that  b  is  to  be  subtracted, 
not  from  a,  but  from  some  other  quantity,  which  is  not  here 
expressed.  Thus  —b  may  represent  the  loss,  which  is  to  be 
subtracted  from  the  stock  in  trade.  (Art.  55.)  The  object 
of  the  calculation,  however,  may  not  require  that  the  value 
of  this  stock  should  be  specified.  But  the  loss  is  to  be  con- 
nected with  a  profit  on  some  other  article.  Suppose  the  profit 
is  2000  dollars,  and  the  loss  400.  The  inquiry  then  is,  what 
is  the  value  of  2000  dollars  profit}  when  connected  with 
400  dollars  loss  ? 

The  answer  is,  evidently,  2000—  400,  which  shows  that 
2000  dollars  are  to  be  added  to  the  stock,  and  400  subtracted 
from  it  ;  or,  which  will  amount  to  the  same,  that  the  differ- 
.encc  between  2000  and  400  is  to  be  added  to  the  stock. 

69.(  Quantities  are  added,  then,  by  writing  them  one  after 
another,  without  altering  their  signs;  observing  always  that  a 
quantity,  to  which  no  sign  is  prefixed,  is  to  be  considered 
positive.  (Art.  29.) 

The  sum  of  et-fm,  and  l>  —  8,  and  2h  —  3r/i-f  d,  and  h  —  n, 
end  r+Sm—  y,  is 

G-fm  +  6  —  8+2A—  3m  +  d+h—  n  +  r+3m  —  y. 

70.  It  is  immaterial  in  what  order  the  tejms  are  arranged. 
The  sum  of  a  and  b  and  c  is  either  a+b  +  c,  'or  «-}-c  +  6,  or 
c+b+a.     For  it  evidently  makes  no  difference,   which  of 
the  quantities  is  added  first.     The  sum  of  t>  and  3  and  9,  is 
the  same  as  3  and  9  and  6,  or  9  and  6  and  3. 

And  a-rm—n,  is  the  same  as  a—  n+m.  For  it  is  plainly 
of  no  consequence,  whether  we  first  add  m  to  a,  and  after- 
wards subtract  n;  or  first  subtract  n,  and  then  add  m. 

71.  Though   connecting  quantities  by  their  signs,  is  all 
which  is  essential  to  addition  ;  yet  it  is  desirable  to  make  the 
expression  as  simple  as  may  be,  by  reducing  several  terms  to 
one.     The  amount  of  3a,  and  Qb,  and  4a,  and  56,  is 


But  this  may  be  abridged.     The  first  and  third  terms  mar 
be  brought  into  one;  and  so  may  the  second  and  fourth.-  — 


30  ALGEBRA. 

For  3  limes  a,  and  4  times  a,  make  7  time?;  a.  And  6  timr  ? 
b,  and  5  times  6,  make  1  1  times  I.  The  sum,  when  redu- 
ced, is  therefore 


For  making  the  reduction?  connected  with  addition,  two 
rules  are  given,  adapted  to  the  two  cases,  in  one  of  which, 
the  quantities  and  signs  are  alike,  and  in  the  other,  the  quan- 
tities are  alike,  hut  the  signs  are  unlike.  Like  quantities  arc 
the  same  powers  of  the  same  letters.  (Art.  45.)  But  as  the 
addition  of  powers  and  radical  quantities  will  he  considered, 
in  a  future  section,  the  examples  given  in  this  place,  will  be 
all  of  the  first  power. 

72.  CASE  I.  To  reduce  several  terms  to  one,  when  the  quan- 
tities are  alike  and  the  signs  alike,  add  the  co-efficients,  annex 
the  common  letter  or  letters,  and  prefix  the  common  sign. 

Thus,  to  reduce  36+76,  that  is,  +36  +  76  to  one  term, 
add  the  co-efficients  3  and  7,  to  the  sum  10,  annex  the  com- 
mon letter  6,  and  prefix  the  sign  +.  The  expression  will 
then  be  +106.  That  3  times  any  quantity,  and  7  times  the 
same  quantity,  make  10  times  that  quantity,  needs  no  proof. 
In  the  same  manner 

6e+26c+96c+36c  becomes,  when  reduced,  156c. 

And  3,ry  +  71ry+Ty+2ry=13zy.  See  the  two  first  of  the 
following  examples. 


.be  3.ry  76+   ry  ry  +  3abh  cdxy  -\-3mg 

26c  7xy  86  +  3ry  3ry  +  'abh  2cdxy+  mg 

96c  xy  26  +  2r^  6n/+406A  5cdxy  -\-1mg 

36c  %ry  66  +  5a;y  2rjr+  abh  Tcdxy+Qmg 


1  56c  236  +  1  Ixy  1  ocdxy  +  1  9mg 


The  mode  of  proceeding  will  be  the  same,  if  the  sign;? 
are  negative. 

Thus  —  36c—  6c—  56r,  becomes,  when  reduced,  —  96r. 
And  —  ax—  Sax—  2«jr  =  —  Qax.     Or  thus, 


—  2ab—  my  —  Srtrh— 

—  ab—Smy  —  ach—  bay 

—  5ach  —  7//r/ 


-10a6-12w? 


ADDITION.  si 

73.  It  may  perhaps  be  asked  here,  as  in  art.  68,  what  pro- 
priety there  is,   in  adding  quantities,  to  which  the  negative 
sign  is  prefixed;  a  sign  which  denotes  subtraction^   The  an- 
swer to  this  is,  that  when  the  negative  sign  is  applied  to  seve- 
ral quantities,  it  is  intended  to  indicate  that  these  quantities 
ure  to  be  subtracted,  not  from  each  other,  but  from  some  oth- 
er quantity,  marked  with  the  contrary  sign.      Suppose  that, 
in  estimating  a  man's  property,  the  sum  of  money  in  his  pos- 
session is  marked  +  ,  and  the  debts  which  he  owes  are  mark- 
ed — .   If  .these  debts  are  200,  300,  500  and  700  dollars,  and 
if  a  is  put  for  100;  they  will  together  be  —  2a— 3a— 5a— 7a. 
And  the  several  terms  reduced  to  one,  will  evidently  be  — 17«, 
that  is,  1700  dollars. 

74.  CASE  II.     To  reduce  several  terms  to  one,  when  the 
quantities  are  alike,  but  the  signs  unlike,  take  the  less  co-ef/icient 
from  the  greater;  to  the  difference,  annex  the  common  letter  or 
letters,  and  prefix  the  sign  of  the  greater  co-efficient. 

Thus  instead  of  8a— 6a,  we  may  write  2a. 
And  instead  of  76—26,  we  may  put  56. 
For  the  simple  expression,  in  each  of  these  instances,  is 
equivalent  to  the  compound  one,  for  which  it  is  substituted. 

To    +66       +46         56c         2Jim     —  dy+6m       3h  —  dx 
Add  -46       —66     — 76c     —  9hm        &dy—  m       5h+4dx 


Sum  +26 


75.  Here  again,  it  may  excite  surprise,  that  what  appears 
to  be  subtraction,  should  be  introduced  under  addition.   But 
according  to  what  has  been  observed,  (Art.  66,)  this  subtrac- 
tion is,  strictly  speaking,  no  part  of  the  addition^*-  It  belongs 
to  a  consequent  reduction.      Suppose  66  is  to  be  added  to 
a— 46.     The  sum  is 

a-46+66. 

But  this  expression  may  be  rendered  more  simple.  As  it 
now  stands,  46  is  to  be  subtracted  from  a,  and  66  added. — 
But  the  amount  will  be  the  same,  if,  without  subtracting  any 
thing,  we  add  26,  making  the  whole  «+26.  And  in  all  sim- 
ilar instances,  the  balance  of  two  or  more  quantities,  may 
be  substituted  for  the  quantities  themselves. 

76.  The  co-efficient  of  a  sum  when  reduced  to  one  term, 
may  be  less,  than  either  of  the  co-efficients  of  the  quanti- 
ties which  are  thus  reduced.      la  one  of  the  preceding  ex- 


32  ALGEBRA. 

amples,  66  —  46=26.  Here  2,  the  co-efficient  of  the  single 
term,  is  less  than  6  or  4,  the  co-efficients  of  the  two  terms, 
to  which  the  single  one  is  equal.  The  balance  of  a  book 
account  may  be  less,  than  either  the  debt,  or  the  credit.  It 
may  even  be  nothing.  Hence, 

77.  If  two  equal  quantities  have  contrary  signs,  they  de- 
stroy each  other,  and  may  be   cancelled.     Thus  +6*r-66 
=0:    And  3x15-18=0:  And  76c—76c=0. 

Let  there  be  any  two  quantities  whatever,  of  swhich  a  is 
the  greater,  and  b  the  less. 

Their  sum  will  be       a+b  ' 
And  their  difference-  a—  b 

The  sum  and  difference  added,  will  be  2«-f-0,  or  simply 
2a.  That  is,  if  the  sum  and  difference,  of  any  two  quantities 
be  added  together,  the  whole  will  be  twice  the  greater  quan- 
tity. This  is  one  instance,  among;  multitudes,  of  the  rapidity 
with  which  general  truths  are  discovered  and  demonstrated 
in  algebra.  (Art.  23.) 

78.  If  several  positive,  and  several  negative  quantities  are 
to  be  reduced  to  one  term  ;   first  reduce  those  which  are 
positive,  next  those  which  are  negative,  and  then  take  the 
difference  of  the  co-efficients,  of  the  two  terms  thus  found. 

Ex.  1.  Reduce  136  +  66+6  —  46  —  56  —  76,  to  one  term. 

By  art.  72,  136+66  +  b-     206 
And  -46-56-76  =  -166 

_  r->; 

By  art.  74        206  —  166=       46,  which  is  the  val- 
ue of  all  the  given  quantities,  taken  together. 

Ex.2.  Reduce  3xy—  xy+2xy  —  Ixy+bxy—  Qxy+lxy  —  Gxy. 
The  positive  terms  are  3xy  The  negative  terms  are  —  xy 


And  their  sum  is       16:cy 
Then 


ADDITION.  33 

Ex.  3.  Sad—  Gad+ad+lad—  2ad+9ad—  Sad—  4a<*=Q. 

4.  2aiw—  •  abm-{-  l(ibm  — 

5.  ary— 


79.  If  the  letters,  in  the  several  terms  to  be  added,  are 
different,  they  can  only  be  placed  after  each  other,  with 
their  proper  signs.     They  cannot  be  united  in  one  simple 
term.      If  4&,  and  —  6y,  and  3x.,  and  17  'h,  and  —  5d,  and  6, 
be  added;-  their  sum  will  be 

Ab-Sy+3x+nh-5d+6.  (Art.  69.) 
Different  letters  can  no  more  be  united  in  the  same  term, 
than  dollars  and  guineas  can  be  added,  so  as  to  make  a  sin- 
gle sum.  Six  guineas  and  4  dollars  are  neither  ten  guineas 
nor  ten  dollars.  Seven  hundred,  and  five  dozen  are  neither 
12  hundred  nor  12  dozen.  But,  in  such  cases,  the  algebraic 
signs  serve  to  show  how  the  different  quantities  stand  related 
to  each  other  ;  and  to  indicate  future  operations,  which  are 
to  be  performed,  whenever  the  letters  are  converted  into 
numbers.  In  the  expression  a  +6,  the  two  terms  can  not 
be  united  in  one.  But  if  a  stands  for  1.5,  and  if,  in  the 
course  of  a  calculation,  this  number  is  restored  ;  then  a+6 
will  become  15+6?  which  is  equivalent  to  the  single  term  21. 
In  the  same  manner,  a—  6  becomes  15—6,  which  is  equal  to 
9.  The  signs  keep  in  view  the  relations  of  the  quantities, 
till  an  opportunity  occurs  of  reducing  several  terms  to  one. 

80.  WheH  the   quantities  to  be   added  contain  several 
terms  which  are  alike,  and  several  which  are  unlike,  it  will 
be  convenient  to  arrange  them  in  such  a  manner,  that  the 
similar  terms  may  stand  one  under  the  other. 

To        3bc—6d+2b—3y  i     These  maybe  arranged  thus. 

Add  —  3bc+x—  3d+bg    >      3bc—  6d-f2&  —  3y 

And      2flH-y+3ff+J       )  —  3bc—3d  +  x+bg 

2d      •  +  y+3x>         +  b 


The  sum  will  be  —  7d+26—  2,y+kc+bg+b. 

In  the  first  term,  3bc  is  balanced  by  —  3bc,  so  that  this 
term  disappears  in  the  general  amount.     (Art.  77.) 


&  ALGEBRA. 

EXAMPLES. 

1.  Add  and  reduce  ab+8tocd—  3  and  5«6—  4m  -f  2. 
The  sum  is  606  +  7  -fed—  Am. 

2.  Addx-fSy—  dx,  to  1—x—S+hm. 
Ans.     3y-odx—  l+hm. 

3.  Add  aim—  3x-f  6m,  to  y—  z-f  7,  and  5x 
Ans. 


4.  Add  3afft+6—7o^—  8,  to  lOxy—  9-f  5ow?, 

Ans. 

5.  Add  6ahy+  Id—  1-fmccy,  to  3<r%—  7<f-H7~ 
Ans. 


6.  Add  7a<Z—  A+8zy—  orf,  to 
Ans, 


SECTION  III 


SUBTRACTION. 

. 

4  1  DDITION  is  bringing  quantities  together,  to 

ART.  bl.^V  find  their  amount.  On  the  contrary,  SUB- 
TRACTION is  finding  the  DIFFERENCE  of  two  quantities,  or  sets 
of  quantities. 

Particular  rules  might  be  given,  for  the  several  cases  in 
subtraction.  But  it  is  more  convenient  to  have  one  general 
rule,  founded  on  the  principle,  that  taking  away  a,  positive 
quantity,  from  an  algebraic  expression,  is  the  same  in  effect, 
as  annexing  an  equal  negative  quantity  ;  and  taking  away  a 
negative  quantity  is  the  same,  as  annexing  an  equal  positive 
one. 

Suppose  -{-b  is  to  be-subtracted  from  a  +6. 

Taking  away  +6,  from  a  +6,  leaves  a. 

And  annexing  —6,  to  a+6,  gives  cr-j-ft—J. 

But  by  axiom  5th,  a+b  —  b  is  equal  to  a. 

That  is,  taking  away  &  positive  term,  from  an  algebraic  ex- 
pression, is  the  same  in  efiect,  as  annexing  an  equal  negative 
term. 

Again,  suppose  —  b  is  to  be  subtracted  from         a—b 

Taking  away  —b,  from  a  —  6,  leaves  a 


And  annexing  +6,  to  a—b,  gives 


a— 


But  a—  b  +  b  is  equal  to 

That  is,  talcing  away  a  negative  term,  is  equivalent  to  an- 
nexing a  positive  one.  If  an  estate  is  encumbered  with  a 
debt  ;  to  cancel  this  debt,  is  to  add  GO  much  to  the  value  of 
the  estate.  Subtracting  an  item  from  one  side  cf  a  book-ac- 
count, will  produce  the  same  alteration  in  the  balance,  as 
adding  an  equal  sum  to  the  opposite  side. 

To  place  this  in  another  point  of  view. 
If  TO  is  added  to  b,  the  sum  is  by  the  notation,      &-f  m  > 
But  if  m  is  subtracted  from  6,  the  remainder  is     b  —  m  .  ) 
So  if  m  and  h  are  each  added  to  b,  the  sum  is         b  -\-m-\-h  > 
T?ut  if  wandAare  each  subtracted  from6,therem'dr  is  b—  m~-h  ) 


jo  ALGEBRA. 

The  only  difference  then  between  adding  a  positive  quan- 
tity and  subtracting  it,  is,  that  the  sign  is  changed  from  + 
to  —  . 

Again,  if  m—  n  is  subtracted  from  b,  the  remainder  is, 

b—  wi+n 

For  the  less  the  quantity  subtracted,  the  greater  will  be  the 
remainder.  But  in  the  expression  m—  n,  m  is  diminished  by 
n  ;  therefore,  6  —  m  must  be  increased  by  n;  so  as  to  become 
b—  m-\-n:  that  is,  m—  n  is  subtracted  frem  6,  by  changing 
-\-m  into  —  m,  and  —  n  into  +n,  and  then  writing  them  after 
6,  as  in  addition.  The  explanation  will  be  the  same,  if 
there  are  several  quantities  which  have  the  negative  sign. 
Hence, 

82.  To  perform  subtraction  in  algebra,  change  the  signs  of 
all  the  quantities  to  be  subtracted,  or  suppose  them  to  be  changed, 
from  +  to  —  ,  or  from  —  to   +  ,  and  then  proceed  as  in  addi- 

tion. 

The  signs  are  to  be  changed,  in  the  subtrahend  only.  — 
Those  in  the  minuend  are  not  to  be  altered.  Although  the 
rule  here  given  is  adapted  to-  every  case  of  subtraction  ;  yet 
there  may  be  an  advantage  in  giving  some  of  the  examples 
in  distinct  classes. 

83.  In  the  first  place,  the  signs  may  be  alike,  and  the  min- 
uend greater  than  the  subtrahend. 


From          +28       166       14da       -28       -166       - 
Subtract     +16       126         Gda       -16       -126        -  6da 


Difference  +12        46         Qda       -12         -46          -Sda 

Here,  in  the  first  example,  the  +  before  16  is  supposed 
to  be  changed  into  — ,  and  then,  the  signs  being  unlike,  the 
two  terms  are  brought  into  one,  by  the  second  case  of  re- 
duction in  addition.  (Art.  74.)  The  two  next  examples  are 
subtracted  in  the  same  way.  In  the  three  last,  the  —  in  the 
subtrahend,  is  supposed  to  be  changed  into  +.  It  may  be 
Well  for  the  learner,  at  first,  to  write  out  the  examples;  and 
actually  to  change  the  signs,  instead  of  merely  conceiving 
them  to  be  changed.  When  he  has  become  familiar  with 
the  operation,  he  can  save  himself  the  trouble  of  transcrib- 
ing- 


37 

84.  In  the  second  place,  the  signs  may  be  alike,  and  the 
minuend  less  than  the  subtrahend. 

From  +  166  *    126          Gda         -16         -i$b        -  Gda 
Sub.    +286       166         Uda         -28         -166        -Uda 


Dif.     -12;-    -46       -Sda         +12  46  Qda 

The  same  quantities  are  given  here,  as  in  the  preceding 
article,  for  the  purpose  of  comparing  them  together.  But 
the  minuend  and  subtrahend  are  made  to  change  places. 
The  mode  of  subtracting  is  the  same.  In  this  class  a  greater 
quantity  is  taken  from  a  7m:  in  the  preceding,  a  less  from  a 
greater.  By  comparing  them,  it  will  be  seen,  that  there  is 
no  difference  in  the  answers,  except  that  the  signs  are  oppo- 
site. Thus  166—126  is  the  same  as  126-166,  except  that 
one  is  +46,  and  the  other  —46:  That  is,  a  greater  quantity 
subtracted  from  a  less,  gives  the  same  result,  as  a  less  sub- 
tracted from  a  greater,  except  that  the  one  is  positive  and 
the  other  negative.  See  art.  58  and  59. 

85.  In  the  third  place,  the  signs  may  be  unlike. 

From  +28       +166       +  14<7a       -28       -166       -14<fa 
Sub.    —16       -126       -  6oto»     +16       +126       +  Qda 


Dif.      +44      +  286        •  2.0da       -44       -28         - 

From  these  examples,  it  will  be  seen  that  the  difference 
between  a  positive  and  a  negative  quantity,  may  be  greater 
than  either  of  the  two  quantities.  In  the  first  example,  44 
the  difference  is  greater,  than  28  the  minuend,  or  16  the 
subtrahend.  In  a  thermometer,  the  difference  between  23 
degrees  above  cypher,  and  16  below,  is  44  degrees.  The 
difference  between  gaining  1000  dollars  in  trade  and  losing 
500,  is  equivalent  to  1500  dollars. 

86.  Subtraction  may  be  proved,  as  in  arithmetic,  by  ad- 
ding the  remainder  to  the  subtrahend.  The  sum  ought  to 
be  equal  to  the  minuend,  upon  the  obvious  principle,  that 
the  difference  of  two  quantities  added  to  one  of  them,  is 
equal  to  the  other.  This  serves  not  only  to  correct  any 
particular  errour,  but  to  verify  the  general  rule. 


38  ALGEBRA. 

From    3abm—  xy     —  ll+Aax          ax+  1b         3ah+axy 
Sub.  —  labm+6xy     —20—  ax     —Aax+I5b      —Tah+axy 


Rern.  lOabm—  Ixy  5ax—  8b 

87.  \Vhen  there  are  several  terms  alike,  they  may  be  redu- 
ced as  in  addition. 

1.  From  ab,  subtract  3am+am+lam+2am+6am. 

Ans.  ab  —  2am—  am—  1am—  2am—  Qam—ub  —  19am.  (Art.72.) 

2.  Fromy,  subtract  —a—  a—  a—  a. 
Ans.  y+a+a+«-f  «=y+4a. 

3.  From  ax—  bc+3ax4-1bc,  subtract  46c—  2ax  +  6c-f  4or. 
Ans.     ax  —  be  +  3ax  +  76c  —  45c  +  2aa?  —  be  —  4ar  =  2a.r  +  ic. 
(Art.  78.) 


4.  Fromad+3f?c—  6ar,  subtract  3ad  4-  7&r—  rfc-f-  a  J. 
Ans. 

88.  When  the  letters  in  the  minuend  are  different  from 
those  in  the  subtrahend,  the  latter  are  subtracted,  by  first 
changing  the  signs,  and  thten  placing  the  several  terms  one 
after  another,  as  in  addition.  (Art.  79.) 

1.  From  3a5  +  8—  my+dh,  subtract  x—  dr+  4Ay—  Imx. 
Ans.  3o6  +  8—  my+dh—  -x+dr—  &hy+bmx. 

2.     IZad+xy+d—  (lad—  xy  +  d+hm—  ry)= 
hm+ry. 


3.  7a6c—  Q+1xJ~3abc—  8— 

4. 

5.  6a?« 


- 


SECTION    IV. 

. 

'MULTIPLICATION* 


A  fiQ  T^  addition,  one  quantity  is  connected  with  a»~ 
ART.  by.  X  Other.  It  is  frequently  the  case,  that  the  quan- 
tities brought  together  are  equal;  that  is,  a  quantity  is  ad- 
ded to  itself. 

As  3+3=6  3+3+3+3=12 

3+3+3=9  3+3+3+3+3  =  15,  &c.  . 


This  repeated  addition  of  a  quantity  to  itself,  is  what  was, 
originally,  called  multiplication.  But  the  term,  as  it  is  now 
used,  has  a  more  extensive  signification.  We  have  frequent 
occasion  to  repeat,  not  only  th<P  whole  of  a  quantity,  but  a 
certain  portion  of  it.  If  the  stock  of  an  incorporated  com- 
pany is  divided  into  shares,  one  man  may  own  ten  of  them, 
another  five,  and  another  a  part  only  of  a  share,  say  two 
fifths.  When  a  dividend  is  made,  of  a  certain  sum  on  a 
share,  the  first  is  entitled  to  ten  times  this  sum,  the  second  to 
jive  times,  and  the  third  to  only  two  fifths  of  it.  As  the  ap- 
portioning of  the  dividend,  in  each  of  these  instances,  is 
upon  the  same  principle,  it  is  called  multiplication  in  the 
last,  as  well  as  in  the  two  first. 

Again,  suppose  a  man  is  obligated  to  pay  an  annuity  of  100 
^dollars  a  year.  As  this  is  to  be  subtracted  from  his  estate,  it 
•may  be  represented  by  —  a.  And  as  it  is  to  be  subtracted 
year  after  year,  it  will  become,  in  four  years,  —  a—  a  —  a—  a 
=  —  4a.  This  repeated  subtraction  is  also  called  multiplica- 
tion. According  to  this  -view  of  the  subject  ; 

*  Newton's  Universal  Arithmetic,  p.  4.  Maseres  on  the  Negative 
Sij^n,  Sec.  II.  Camus'  Arithmetic,  Book  II.  Chap.  3.  Euler's  Alge- 
bra, Sec.  I.  and  II.  Chap.  S.  Simpson's  Algebra,  Sec.  IV.  Maclau- 
:i  %  Saui.derson,  Lacroix,  Ludlarn. 


40  ALGEBRA. 

90.  Multiplying  by  a  whole  number  is  taking  the,  multipli- 
tan-l  as  many  rimes,  as  there  are  units  in  the  multiplier.* 
Multiplying  by  1,  is  taking  the  multiplicand  on^,  as  a. 
Multiplying  by  2,  is  taking  the  multiplicand  tunce,  as  a -fa. 
Multiplying  by  3,  is  taking  the  mult'd  three  timcs,*s  a  +  a-f-  a,&c. 

Multiplying  by  a  FRACTION  is  taking  a  certain  PORTION  of 
the  multiplicand  as  many  times,  as  there  are  like  portions  of  an 
unit  in  the  multiplier. 

Multiplying  by  |,  is  taking  |  of  the  mult'd  once,  as  |a. 
Multiplying  by  |,  is  taking  |  of  the  mult'd  twice,  as  -lsa-\-^a. 
Multipl'g  by  f,is  taking  jof  the  mult'd  three  times,zs  $a  -f  lga  +  }a. 

Hence,  if  the  multiplier  is  an  unit,  the  product  is  equal  to 
the  multiplicand :  If  the  multiplier  is  greater  than  an  unit, 
the  product  is  greater  than  the  multiplicand:  And  if  the  mul- 
tiplier is  less  than  an  unit,  the  product  is  less  than  the  multi- 
plicand. 

Multiplication  by  a  NEGATIVE  quantity,  has  the  same  rela- 
tion to  multiplication  by  a  positive  quantity,  which  SUBTRAC- 
TION A«w  to  addition.  In  the  one,  the  sum  of  the  repetitions 
of  the  multiplicand  is  to  be  added,  to  the  other  quantities 
with  which  the  multiplier  is  connected.  In  the  other,  the 
sum  of  these  repetitions  is  to  be  subtracted  from  the  other 
quantities.  This  subtraction  is  performed  at  the  time  of 
multiplying,  by  changing  ttffe  sign  of  the  product.  See  Art. 
107  and  108. 

94.  Every  multiplier  is  to  be  considered  a  number.  We 
sometimes  speak  of  multiplying  by  a  given  weight,  or  meas- 
ure, a  sum  of  money,  &c.  But  this  is  abbreviated  language. 
Jf  construed  literally,  it  is  absurd.  Multiplying  is  taking 
either  the  whole  or  a  part  of  a  quantity,  a  certain  number  of 
times.  To  say  that  one  quantity  is  repeated  as  many  times, 
as  another  is  heavy,  is  nonsense.  But  if  a  part  of  the  weight 
of  a  body  be  fixed  upon  as  an  unit,  a  quantity  may  be  mul- 
tiplied by  a  number  equal  to  the  number  of  these  parts  con- 
tained in  the  body.  If  a  diamond  is  sold  by  weight,  a  par- 
ticular price  may  be  agreed  upon  for  each  grain.  A  grain  is 
here  the  unit ;  and  it  is  evident  that  the  value  of  the  dia- 
mond, is  equal  to  the  given  price  repeated  as  many  times, 
as  there  are  grains  in  the  whole  weight.  We  say  concisely 
that  the  price  is  multiplied  by  the  weight;  meaning  that  it 
is  multiplied  by  a  number  equal  to  the  number  of  grains  in 
the  weight.  In  a  similar  manner,  any  quantity  whatever 

*  Sec  note  A.  at  the  end. 


MULTIPLICATION.  41 

may  br  supposed  to  be  made  up  of  parts,  each  being  con- 
sidered a  -unit,  and  any  number  of  these  may  become  a 
multiplier. 

92.  As  multiplying  is  taking  the  whole  or  a  part  of  a 
quantity  a  certain  number  of  times,  it  is  evident  that  the 
•product  must  be  of  the  same  nature  as  the  multiplicand. 

If  the  multiplicand  is  an  abstract  number;  the  product 
will  be  a  number. 

If  the  multiplicand  is  weight,  the  product  will  be  weight. 
If  the  multiplicand  is  a  line,  the  product  will  be  a  line.  Re- 
peating a  quantity  does  not  alter  its  nature.  It  is  frequently 
said,  that  the  product  of  two  lines  is  a  surface,  and  that  the 
product  of  three  lines  is  a  solid.  But  these  are  abridged  ex- 
pressions, which  if  interpreted  literally  are  not  correct*  See 
the  section  on  the  application  of  Algebra  to  Geometry, 

93.  The  multiplication  of  fractions  will  be  the  subject  of 
a  future  section.      We  have  first  to  attend  to  multiplication 
by  positive  whole  numbers.      This,  according  to  the  defini- 
tion (Art.  90)  is  taking  the  multiplicand  as  many  times,  as 
there  arc  units  in  the  multiplier.    Suppose  a  is  to  be  multiplied 
by  b,  and  that  b  stands  for  3.      There  are,  then,  three  units 
in  the   multiplier  b.      The  multiplicand  must  therefore  be 
taken  three  times ;  thus, 


The  amount  is  3a,  that  is,  3  times  a ',  which,  if  b  stands 
for  3,  is  the  same  as  b  x  a  or  ba.  (Art.  40.)  Or  thus,  a  +  «-f« 
=3«,  which  is  the  same  as  ba<  So  that,  multiplying  two 
letters  together  is  nothing  more,  than  writing  them  one  after  the 
other,  either  with,  or  without,  the  sign  of  multiplication  be- 
tween them.  Thus  b  multiplied  into  c,  is  b  x  c,  or  be.  And 
x  into  y,  is  x  x  y,  or  x,y,  or  xy. 

94.  If  more  than  two  letters  arc  to  be  multiplied,  they 
must  be  connected  in  the  same  manner.      Thus  a  into  l>  anil 
c,  is  eba.     For  by  the  last  article,  n  into  b,  is  ba.     This  pro- 
duct is  now  to  be  multiplied  into  c.     If  c  stands  for  5,  then 
ba  is  to  be  taken  five  times,  thus, 

ba-\-ba-\-ba  +  ba+ba=5ba,  or  cba. 

The  same  explanation  may  be  applied  to  any  number  of 
letters.  Thus  am  into  xy,  is  cwixy.  And  bh  into  mrx,  is- 
bhmrx. 

95.  It  is  immaterial  ;';?  what  order  the  letters  arc  arranged. 

G 


42  ALGEBRA. 

The  product  ba  is  the  same  as  ab.  Three  times  fire  is 
equal  to  five  times  three.  Let  the  number  5  be  represented 
by  as  many  points,  in  a  horizontal  line ;  and  the  number  3, 
by  as  many  points  in  a  perpendicular  line. 


Here  it  is  evident  that  the  whole  number  of  points  is  equal, 
either  to  the  number  in  the  horizontal  row  three  times  repeat- 
ed, or  to  the  number  in  the  perpendicular  row  Jive  times  re- 
peated; that  is  to  5x3,  or  3x5.  This  explanation  may  be 
extended  to  a  series  of  factors  consisting  of  any  numbers 
whatever.  For  the  product  of  two  of  the  factors  may  be 
considered  as  one  number.  This  may  be  placed  before  or 
after  a  third  factor:  the  product  of  three,  before  or  after  a 
fourth,  &x. 

Thus  24=4x6  or  6x4=4x3x2  or  4x2x3  or  2x3x4. 
The  product  of  a,  b,  c,  and  d,  is  abcd>  or  acdb,  or  dcba,  or  bade. 
It  will  generally  be  convenient,  however,  to  place  the  letters 
in  alphabetical  order. 

96.  When  the  letters  have  numerical  CO-EFFICIENTS,  these 
must  be  multiplied  together,  and  prefixed  to  the  product  of  the 
letters. 

Thus  3a  into  2b  is  Gab.  For  if  a  into  b  is  ab,  then  3 
times  a  into  b,  is  evidently  Sab :  and  if,  instead  of  multiply- 
ing by  b,  we  multiply  by  twice  b,  the  product  must  be  twice 
as  great,  that  is  2  X  Sab  or  Gab. 

Mult'y     9a5         1%         Srf/t  2ad  Ibdh         Say 

Into     .    Sxy          2-ra1  my         IShmg          x  Qnix 

Prod.    27abxy  Sdhmy  „  Ibdhx 


97.  If  either  of  the  factors  consists  of  figures  only,  these, 
must  be  multiplied  into  the  co-efficients  and  letters  of  the 
other  factors. 

Thus  Sab  into  4,  i*  12ab.      And  36  into  2r,  is  72r.      And 
24  into  hy,  is  2&hy. 

98.  If  the  multiplicand  is  a  compound  quantity,  each  of  its 
terms  must  be  multiplied  into  the  multiplier.     Thus  b  +  c-\-d 
into  a  is  ab -\-ac-\- ad.      For  the  whole  of  the  multiplicand  is 
to  be  taken  as  many  times,  as  there  are  units  in  the  multi- 


MULTIPLICATION.  43 

pller.      If  then  a  stands  for  3,  the  repetitions  of  the  multi- 
plicand are 

b+c+d 

l+c+d 

b+c+d 


And  their  sum  is      36  +  3c + 3d,  that  is,  ab  +  ac  +  ad 

• 

Mult.     d+2xy          2Ji+m  2hl+l  2hm+3+dr 

Into     3b  7dy  my  4i 

, —      ___—__—  .  -     "  •• 

Prod.  3b d+  Qbxy  /  &  Shlmy + my 


99.  The   preceding  instances  must  not  be  confounded 
with  those  in  which  several  factors  are  connected  by  the 
sign  x ,  or  by  a  point.     In  the  latter  case,  the  multiplier  is 
to  be  written  before  the  other  factors  without  being  repeated, 
The  product  of  b  x  d  into  a,  is  ab  X  </,  and  not  ab  x  ad.    For 
b  x  d  is  bd,  and  this  into  a,  is  abd.  (Art.  94.)    The  expression 
bxd  is  not  to  be  considered,  like  b  +  d,  a  compound  quantity 
consisting  of  two  terms.     Different  terms  are  always  separa- 
ted by  +  or  — .    (Art.  36.)      The  product  of  bxhxmxy 
into  «,  is  axbxhxmxy  or  abhmy.  But  b+h+m+y  into  a, 
is  ab  +  ah  +  am+ay. 

100.  If  both  the  -factors  are  compound  quantities,  each 
term  in  the  multiplier  .must  be  multiplied  into  each  in  the  multi- 
plicand^ 

Thus  a+b  into  c+d  is  ac+ad+  bc+bd. 

For  the  units  in  the  multiplier  a+b  are  equal  to  the  units 
in  a  added  to  the  units  in  b.     Therefore  the  product  produ- 
ced by  a,  must  be  added  to  the  product  produced  by  b. 
The  product  of  c+d  into  a  is  ac+ad   )   A      QR 
The  product  of  c+d  into  b  is  bc  +  bd    $ 
The  product  of  c+d  into  a+b  is  therefore  ae+ad+bc+bd. 

Mult.  Zx+d  4ay+2b        a  +  1 

Into     2a+hm  3c   +rx      3oc-f4 


Prod.  6ax+2ad+3hmx+dhm  3ao?+3.r-f4a+4 


41  ALGEBRA. 

Mult.  2Ji  +  7  into  6i/+ 1.     Prod.  I2dh+*2d+2h  +  7. 
Mult,  dy  +  rx+h  into  Gjn+4+7y.     Prod. 
Mult.  7  +  66  +  ad  into  3r + 4 + 2h.     Prod. 

101.  When  several  terms  in  the  product  are  alike,  it  will 
be  expedient  to  set  one  under  tke  other,  and  then  to  unite 
them,  by  the. rules  for  reduction  in  addition. 

a+  y+l 
36+2z  +  7 


Mult.  6  +  a                 b  +  c  +  2, 
Into     6  +  a                 6  +  c+3 

bb  +  ab 

bb  +  bc  +  2b 
be         + 
+  36 

cc+2c 
+  3c+G 

Prod. 


Mult.  3a+d+4  into  2a+3rf+l.     Prod. 
Mult.  6+cd+2  into  36+4cd4-7.     Prod. 

102.  Here,  as  in  Art.  99,  care  must  be  taken  not  to  con- 
found terms  with  factors. 

The  product  of  a  x  b  into  c  X  rf,  is  a  X  6  X  c  X  d,  or  rt&c<7. 
But  the  product  of  a-fJ  into  c+rf,  is  ac+ad+bc  +  bd. 
The  product  of  36+2c  into  Axm,  is  obhm  -\-2chm. 
The  product  of  rtX&Xc  into  /i  +  4y,  is  abch+Aabcy. 

103.  It  will  be  easy  to  see  that  when  the  multiplier  and 
multiplicand  consist  of  any  quantity  repeated  as  a  factor,  this 
factor  will  be  repeated  in  the  product,  as  many  times  as  in 
jhe  multiplier  and  multiplicaad  together. 

Mult,  a  X  a  x  «         Here  a  is  repeated  three  times  as  a  factor. 
Juto     ax  a  Here  it  is  repeated  twice. 

Prod,  axaxaxaxa  Here  it  is  repeated  five  times. 


The  product  of  bbbb  into  bbb,  is  bbbbbbb. 

The  product  of  2.x  x  3r  x  4x  into  5z  x  6a?,  is  2r  x  3x  x  4r 
X  5x  x  6#. 

104.  But  the  numeral  co-efficients  of  several  fellow-factors 
njay  be  brought  together  by  multiplication. 

Thus  2a x  3b  into  4«  x  56  is  2a  x  36  x  4« x  5£,  or  l2Qaabb. 


MULTIPLICATION.  45 

For  the  co-efficients  are  factors^  (Art.  41,)  and  it  is* imma- 
terial in  what  order  these  are  arranged.  (Art.  95.)     So  that 
2rtx3&x4ax5&=2x3x4x5xaxaX&X&  =  IZQaabb. 
The  product  of  3a  X  Abh  into  5m  x  6y,  is  360  abhmy. 
The  product  of  4&x6d  into  2r-f-l,  is  ASbdx+Zlbd. 

105.  The  examples  in  multiplication  thus  far  have  been 
confined  to  positive  quantities.      It  will  now  be  necessary  to 
consider,,  in  what  manner  the  result  will  be  affected,  by  mul- 
tiplying positive  and  negative  quantities  together.     We  shall 
find, 

That    +  into  +  produces  -f 

—  into  -f 
4-  into  — 

—  into  —  -f 

All  these  may  be  comprised  in  one  general  rule,  which  it 
will  be  important  to  have  always  familiar.  If  the  signs  of  the 
factors  are  ALIKE,  the  sign  of  the  product  will  be  affirmative  ; 
but  if  the  signs  of  the  factors  are  UNLIKE,  the  sign  of  the  pro- 
duct will  be  negative. 

106.  The  first  case,  that  of  4-  into  +  »  needs  no  farther 
illustration.     The  second  is  —  into  +,  that  is,  the  multipli- 
cand is  negative,  and  the  multiplier  positive.      Here  — a  in- 
to +4  is  — 4a.     For  the  repetitions  of  the  multiplicand  are, 

— a  — a— a— a=—  4a. 

Mult,     b— 3a  2a— m      h  —  3^—4  a— 2— Id— x 

Into     6y  3h+x     2y  3b+h 

Prod.Qby—lSay  2hy—6dy—8y 


107.  In  the  two  preceding  cases,  the  affirmative  sign  prefix- 
ed to  the  multiplier  shows,  that  the  repetitions  of  the  multi- 
plicand are  to  be  added,  to  the  other  quantities  with  which 
the  multiplier  is  connected.  But  in  the  two  remaining  cases, 
the  negative  sign  prefixed  to  the  multiplier,  indicates  that  the 
sum  of  the  repetitions  of  the  multiplicand  are  to  be  subtract- 
ed from  the  other  quantities.  (Art.  90.)  And  this  subtrac- 
tion is  performed,  at  the  time  of  multiplying,  by  making  the 
sign  of  the  product  opposite  to  that  of  the  multiplicand. 
Thus  -f#  into  —4,  is  —  4a.  For  the  repetitions  of  the  mul- 
tiplicand are, 


But  this  suin  is  to  be  subtracted,  from  the  other  quantities 


4fl  ALGEBRA. 

with  which  the  multiplier  is  connected.  It  will  then  become 
~4n.  (Art.  82.) 

Thus,  in  the  expression  b  —  (4x  a),  it  is  manifest  that  4x  a 
is  to  be  subtracted  from  b.  Now  4xo  is4rt,  that  is,  +  4a. 
But,  to  subtract  this  from  b,  the  sign  -f  must  be  changed  in- 
to —  .  So  that  b  —  (4Xff)  is  b—  4a.  And  «x  —4  is  there- 
fore —  4a. 

Again,  suppose  the  multiplicand  is  a,  and  the  multiplier 
(6—4).  As  (6—4)  is  equal  to  2,  the  product  will  be  equal  to 
2a.  This  is  less  than  the  product  of  6  into  «.  To  obtain 
then  the  product  of  the  compound  multiplier  (6—4)  into  a, 
we  must  subtract  the  product  of  the  negative  part,  from 
that  of  the  positive  part. 


And  the  prodr  6a—  4a,  is  the  same  as  the  product     2a. 
Therefore  a  into  —4,  is—  4a. 

But  if  the  multiplier  had  been  (6+4),  the  two  products 
must  have  been  added. 

Multiplying        a  >  .   ^  ^  <  Multiplying  a 

Into  6+4)  (Into  10 

And  the  prod.    6a+4a  is  the  same  as  the  product  10a 

This  shows  at  once  the  difference  between  multiplying  by 
a  positive  factor,  and  multiplying  by  a  negative  one.  In  the 
former  case,  the  sum  of  the  repetitions  of  the  multiplicand 
is  to  be  added  to,  in  the  latter,  subtracted  from,  the  other 
quantities,  with  which  the  multiplier  is  connected.  For  eve- 
ry negative  quantity  must  be  supposed  to  have  a  reference 
to  some  other  which  is  positive  ;  though  the  two  may  not  al- 
ways stand  in  connection,  when  the  multiplication  is  to  be 
performed. 

Mult.  «+6  3dy+hx+2        3A  +3 

Into     b—  x  mi  —  ab  ad—  6 


Prod,  ab+bb-ax-bx  3adh  +  lad—lSh-l8 

108.    If  two  negatives  be  multiplied  together,  the  product 


MULTIPLICATION.  41 

will  be  affirmative  :.— 4x  — a  =  -f-4«-  In  this  case,  as  in  the 
preceding,  the  repetitions  of  the  multiplicand  are  to  be  sub- 
tracted, because  the  multiplier  has  the  negative  sign.  These 
repetitions,  if  the  multiplicand  is  —a  and  the  multiplier— 4, 
are  _a_a_a_rt=r— 4a.  But  this  is  to  be  subtracted  by 
changing  the  sign.  It  then  becomes  -fkr. 

Suppose  —a  is  multiplied  into  (6—4).  As  6— 4=2,  the 
product  is  evidently,  tmce  the  multiplicand,  that  is  —  2cr. 
But  if  we  multiply  —a into  6  and  4  separately;  —a  into  6 
is  — 6a,  and  —a  into  4  is  — 4a.  (Art.  106.)  As,  in  the  mul- 
tiplier, 4  is  to  be  subtracted  from  6 ;  so,  in  the  product,  —  4« 
must  be  subtracted  from  —  6«.  Now  —  4a  becomes  by  sub- 
traction +4a.  The  whole  product  then  is  —  6a+Aa,  which 
is  equal  to  2g.  Or  thus, 

Multiplying     -a  >  .     }  C  Multiplying  -a 

Into  6  —4  5  (.  Into  2 

And  the  prod.     — 6a+4«,  is  equal  to  the  product       — 2a. 

In  Double  Fellowship  in  arithmetic,  each  man's  stock  is 
to  be  multiplied  into  the  time  for  which  it  is  employed.  Sup- 
pose there  are  two  partners  A  and  B ;  that  B's  stock  is  300 
dollars  less  than  A's;  and  that  the  time  of  the  former  is  two 
years  less,  than  that  of  the  latter :  then  , 

If  A's  share  is  equal  to  c ;  B's  share  will  equal  c— 300  > 
Andlf  A's  time  is  equal  to  d;  B's  time  will  equal   d— 2      y  . 

Multiplying  c— 300 

Into  d-2 

' 


The  product  will  be       cd—  300d—  2r+600. 

Here  the  two  first  terms  are  obtained,  by,  multiplying 
(c—  300)  into  d;  that  is,  B's  stock  into  the  whole  time  repre- 
sented by  d.  But  this  time  is  two  years  too  much.  The  pro- 
duct is  therefore  too  great.  It  ought  to  be  diminished,  by 
the  product  of  the  stock  (c—  300)  into  2.  The  whole  pro- 
duct will  then  be 


. 

Here  2c—  600  is  subtracted  by  changing  the  signs  (Art.  88.) 
so  that  —  300  x  —2  is  +600.  On  the  same  principle,  it  is 
necessary  to  change  the  signs  of  every  term  in  a  compound 
quantity  which  is  multiplied  by  a  negative  factor. 


48  ALGEBRA. 

It  is  often  considered  a  great  mystery,  that  the  product  of 
two  negatives  should  be  affirmative.  Cut  it  amounts  to  no- 
thing more  than  this,  that  the  subtraction  of  a  negative 
quantity,  is  equivalent  to  the  addition  of  an  affirmative, 
(Art.  81,)  and,  therefore,  that  the  repented  subtraction  of  a 
negative,  is  equivalent  to  a  repealed  addition  of  an  affirma- 
tive. Taking  off  from  a  man's  hands  a  debt  of  ten  dollars 
every  month,  is  adding  ten  dollars  a  month  to  the  value  of 
his  property. 

Mult,     o— 4  3d—hy—2x   3«y— b 

Into     36-6  46-7  &r-l 


Prod.  3«6  — 126— 6a+24  ISaxy  — Gbx  — 3ay+b 


Multiply  Sad— ah  —  7  into  4— dy  — Jir. 
Multiply  C2hy+3m— I  into  4rf  —  2x+3. 

109.  As  a  negative  multiplier  changes  the  sign  of  the 
quantity  which   it  multiplies;  if  there  are  severed  negative 
factors  to  be  multiplied  together, 

The  two  first  will  make  the  product  positive; 
The  third  will  make  it  negative; 
'  The  fourth  will  make  it  positive,  Sic. 

Thus  —a  And  —abc 

Into    —I  Into  —  d 

[factors.  

Gives  +  ab,  the  prod,  of  two  Gives  +  abed,  four  fsucCf. 
This  into     —  c  This  into  —  e 

Gives  —  abc,  three  factors.      Gives  —abcde,Jire  fact's. 

That  is, 

The  product  of  any  even  number  of  negative  factors  is 
positive;  but  the  product  of  any  odd  number  of  negative 
factors  is  negative. 

Thus  —ax  —a  —  aa  And  — ax  —ax  —ax  —a=aaan 

—ax  —ax  —a—-aaa    —ax  —ax  —ax  —  ax-a=z-aaa«a. 

The  product  of  several  factors  which  are  all  positive,  i* 
invariably  positive. 

110.  Positive  and  negative  terms  may  frequently 


MULTIPLICATION.  49 

each  other,  so  as  to  disappear  in  the  product.  (Art.  77.)     A 
star  is  sometimes  put  in  the  place  of  the  deficient  term. 

Mult,  a— b  «  mm—yy        aa+ab+bb 

Into     a+b  mm+yy          a—b 


aa—ab  aaa+aab+abb 

-\-ab-bb  —aab—abb—bbb 


Prod.aa    *    -bb  aaa      *        *    -bbb 


111.  For  many  purposes,  it  is  sufficient  merely  to  indicate 
the  multiplication  of  compound  quantities,  without  actually 
multiplying  the  several  terms.     Thus  the  product  of 

a  +  b  +  c  into  h+m+y,  is  (a+b  +  c)  x  (h+m+y).   (Art.  40.) 
The  product  of 

«-f  TO  into  h+x  and  d+y,  is  (a+  m)  x(h+x)x(d+y). 
By  this  method  of  representing  multiplication,  an  important 
advantage  is  often  gained,  in  preserving  the  factors  distinct 
from  each  other. 

When  the  several  terms  are  multiplied  in  form,  the  ex- 
pression is  said  to  be  expanded.  Thus  [Art.  100,, 

(a-\-b)x(c+d)  becomes  when  expanded  ac-^-ad-^- bc+bd, 

112.  With  a  given  multiplicand,  the  less  the  multiplier, 
the  less  will  be  the  product.     If  then  the  multiplier  be  redu- 
ced to  nothing,  the  product  will  be  nothing.     Thus  a  x  0=0. 
And  if  0  be  one  of  any  number  of  fellow-factors,  the  product 
of  the  whole  will  be  nothing. 

Thus  ab  x  c  x  3d  x  0=3abcd  x  0  =0. 
And  (a+b)  x  (c+d)  x  (A— TO)  x  0=0. 

113.  Although,  for  the  sake  of  illustrating  the  different 
points  in  multiplication,  the  subject  has  been  drawn  out  into 
a  considerable  number  of  particulars  ;  yet  it  will  scarcely  be 
necessary  for  the  learner,  after  he  has  become  familiar  with 
the  examples,  to  burden  his  memory  with  any  thing  more 
than  the  following  general  rule. 

Multiply  the  letters  and  co-efficients  of  each  term  in  the  mul- 
tiplicand, into  the  letters  and  co-efficients  of  each  term  in  the 
multiplier ;  and  prefix,  to  each  term  of  the  product,  the  sign 
required  by  the  principle,  that  like  signs  produce  +5  and  dif- 
ferent signs  —'. 


50  ALGEBRA. 

Mult.  a+36-2into4a— 6&— 4. 
Mult.  Aab  x  a?  X  2  into  3my — 1 +A. 
Mult.  (7aA — y)  x  4  into  4r  x  3  X  5  x  rf. 
Mult.  (6a6— M+l)x2into  (8+4^- 
Mult.  3ay+y— 4+A  into  (d+x)  x 


SECTION    V. 


DIVISION. 


A  111  T^  multiplication?  we  have  two  factors  giver*, 
and  are  required  to  find  their  product.  By 
multiplying  the  factors  4  and  6,  we  obtain  the  product  24. 
But  it  is  frequently  necessary  to  reverse  this  process.  The 
number  24,  and  one  of  the  factors,  may  be  given,  to  enable 
us  to  find  the  other.  The  operation  by  which  this  is  effect- 
ed is  called  Division.  We  obtain  the  number  4,  by  dividing 
24 by  6.  The  quantity  to  be  divided  is  called  the  dividend; 
the  given  factor,  the  divisor ;  and  that  which  is  required,  the 
quotient. 

115.  DIVISION  is  finding  a  quotient,  which  multiplied  into 
the  divisor  will  produce  the  dividend.* 

In  multiplication,  the  multiplier  is  always  ^  number.  (Art. 
91.)  And  the  product  is  a  quantity  of  the  same  kind,  as  the 
multiplicand.  (Art.  92.)  The  product  of  3  rods  into  4,  is 
12  rods.  When  we  come  to  division,  the  product  and  either 
of  the  factors  may  be  given,  to  find  the  other :  that  is, 

The  divisor  may  be  a  number,  and  then  the  quotient  will 
be  a  quantity  of  the  same  kind  as  the  dividend ;  or 

The  divisor  may  be  a  quantity  of  the  same  kind  as  the  div- 
idend ;  and  then  the  quotient  will  be  a  number. 

12  rods  12  rods 


Thus -A — =3  rods.  But  ,  —  >* 

4  3  rods 

12  rods  12  rods 

And  — ^-j — =-i  roc?.  And  ^7 — T=? 

24  24  roas     •* 

In  the  first  case,  the  divisor,  being  a  number,  shows  int» 
how  many  parts  the  dividend  is  to  be  separated ;  and  the  quo- 
tient shows  what  these  parts  are. 

*The  remainder  is  here  supposed  to  be  included  iu  Ihe  quotient* 
-as  is  commonly  the  case  in  algebra. 


52  ALGEBRA. 

If  12  rods  lir  divided  into  4  parts,  each  will  be  3  rods  long. 
And  if  12rodsbe  divided  into  24parts,each  will  be  halfurod  long. 

In  the  other  case,  if  the  divisor  is  less  than  the  dividend, 
the  former  shows  into  what  parts  the  latter  is  to  be  divided ; 
and  the  quotient  shows  how  many  of  these  parts  are  contain- 
ed in  the  dividend.  In  other  words,  division  in  this  case 
consists  in  finding  how  often  one  quantity  is  contained  in  an- 
other. 

A  line  of  3  rods,  is  contained  in  one  of  12  rods,  four  times. 

But  if  the  divisor  is  greater  than  the  dividend,  and  yet  a 
quantity  of  the  same  kind,  the  quotient  shows  what  part  of 
the  divisor  is  equal  to  the  dividend. 

Thus  one  half  of  24  rods  is  equal  to  12  rods. 

116.  As  the  product  of  the  divisor  and  quotient  is  equal  to 
the  dividend,  the  quotient  may  be  found,  by  resolving  the 
dividend  into  two  such  factors,  that  one  of  them  shall  be  the 
divisor.  The  other  will,  of  course,  be  the  quotient. 

Suppose  abd  is  to  be  divided  by  a.  The  factors  a  and 
bd  will  produce  the  dividend.  The  first  of  these,  being  a 
divisor,  may  be  set  aside.  The  other  is  the  quotient.  Hence, 

When  the  divisor  is  found  as  a  factor,  in  the  dividend,  the  di+ 
vision  is  performed,  by  CANCELLING  this  factor. 

Divide  ex        dh 
By       c          d 

Quot.    x 


In  each  of  these  examples,  the  letters  which  are  common 
to  the  divisor  and  dividend,  are  set  aside,  and  the  other  let- 
ters form  the  quotient.  It  will  be  seen  at  once,  that  the 
product  of  the  quotient  and  divisor  is  equal  to  the  dividend. 

117.  If  a  letter  is  repeated  in  the  dividend,  care  must  be 
taken  that  the  factor  rejected  be  only  equal  to  the  divisor. 

Div.     aab       bbx       aadddx      aammyy        aaaxxxh         yyy 
By       <•          b  ad  amy  aaxx  yy 

,Quot.    ab  addx  a-xh 

Jn  such  instances,  it  is  obvious  that  we  are  not  to  reject 


DIVISION.  53 

every  letter  in  the  dividend  which  is  the  same  with  one  in  the 
divisor. 

118.  If  the  dividend  consists  of  any  factors  whatever^,  ex- 
punging one  of  them  is  dividing  by  it. 

Div.  a(b+d)      a(b+d)        (b+x)(c+d)        (b+y)x(d-h)x 
By      a  b  +  d  b+fr  d—h 

Quot.  b+d  a  c+d  (b+y)xx 


In  all  these  instances  the  product  of  the  quotient  and  divi- 
sor is  equal  to  the  dividend  by  Art.  111. 

119.  In  performing  multiplication,  if  the  factors  contain 
numeral  figures,  these  are  multiplied  into  each  other.  (Art. 
96.)  Thus  3a  into  76  is  21  ab.  Now  if  this  process  is  to  be 
reversed,  it  is  evident  that  dividing  the  number  in  the  pro- 
duct, by  the  number  in  one  of  the  factors,  will  give  the  num- 
ber in  the  other  factor.  The  quotient  of  21a6-i-3a  is  76. 
Hence, 

In  division,  if  there  are  numeral  co-efficients  prefixed  to 
the  letters,  the  co-efficient  of  the  dividend  must  be  divided,  by 
the  co-efficient  of  the  divisor. 

Div.     Gab         IQdxy        25dhr       12xy        3&drx         2Qhm 
By      26  4<fo  dh          6  34  m 


Quot.  3a  25r 


120.  When  a  simple  factor  is  multiplied  into  a  compound 
one,  the  former  enters  into  every  term  of  the  latter.  (Art.  98.) 
Thus  a  into  b  +  d,  isab  +  ad.  Such  a  produet  is  easily  resol- 
ved again  into  its  original  factors.* 

Thus  ab  +  ad=ax(b  +  d) 

amh  +  amx + amy =amx(h+x+y) 


Now  if  the  whole  quantity  be  divided  by  one  of  these  fac- 
tors, according  to  Art.  118,  the  quotient  will  be  the  other 
factor. 

Thus  (ab  +  ad)-±a-i+d.      And  (ab +ad}~(l>+ d)=a. 
Hence, 


f4  ALGEBRA.  . 

If  the  divisor  is  contained  in  every  term  of  a  compound  div- 
idend, it  must  be  cancelled  in  each. 

Div.     ab  +  ac        bdh+bdy        aah-\-ay         drx+dhx-\-dxy 
By       a  bd  a  dx 

Quot.  b  +  c  ah+y 


And  if  there  are  co-efficients,  these  must  be  divided,  in 
each  term  also. 


Div.     6ab  +  12ac       10rfry+16J       12hx+8 
By       3o  2eZ  4 


Quot.  26+4c  3/b+2 



121.  On  the  other  hand,  if  a  compound  expression  contain- 
ing any  factor  in  every  term,  be  divided  by  the  other  quantities 
connected  by  their  signs,  the  quotient  icill  be  that  factor.  See 
the  first  part  of  the  preceding  article^ 


Div.    ab  -\-ac-\-  ah    amh  -\-amx  -\-amy 

By       b  +  c  +  h          k+x+y  b  +  2y        m+y 


Quot.  a 


122.  In  division,  as  well  as  in  multiplication,  the  caution 
must  be  observed,  not  to  confound  terms  with  factors.  See 
Arts.  99  and  102. 


Thus  («6+ac)-T-  «=*+«.-  (Art.  120.) 
But    (aby.ac\-±a=aabc-:ra=abc. 
And   (ab-\-ac)+(l>+c)=a.     (Art.  121.) 
But    (abxac)-:r(bxc)=aabc^-bc=:aa. 

123.  In  division  the  same  ruk;  is  to  be  observed  respecting 

the  signs,  as  in  multiplication  ;  that  is,  if  the  divisor  and  div- 

idend are  both  positive,  or  both  negative,  the  quotient  must  be 

^positive  :  if  one  is  positive  and  the  other  negative,  the   quotient 

^must  be  negative.  (Ail.  105.) 

This  is  manifest  from  the  consideration  that  the  product  of 
ike  divisor  and  quotient  must  be  the  same  as  the  dividend. 


DIVISION. 


If  -faX+fc^-foM  f  -\-ab-r-  +b- 

_ax+i  =  —  ab  (     ,          J  -a5-=-+6  = 
6  =  -aH  -ai^--6  = 


—  ax—  6  = 


Div.     «&#         Sa  —  lOcry       3ax—6ay       Gamxdh 
By       —a         —  2a  3a  —2a 

Qiiot.  —  bx         —  4-f-5y  —3mxdh=-3dhm 


124.  ^  Me  letters  of  the  divisor  are  not  to  be  found  in  the 
dividend,  the  division  is  expressed  by  vjriting  the  divisor  under 
the  dividend,  in  the  form  of  a  vulgar  fraction. 

Div.     xy       6hr       d—x       2d—r       d—h+3y        r+x  +  l 
By        a       Ady       —h         —  3r         m—x  d+2h—y 

xy  d—x  d—h+3y 

~  - 


This  is  a  method  of  denoting  division,  rather  than  an  actu- 
al performing  of  the  operation.  But  the  purposes  of  division 
may  frequently  be  answered,  by  these  fractional  expressions. 
As  they  are  of  the  same  nature  with  other  vulgar  fractions, 
they  may  be  added,  subtracted,  multiplied,  &c.  See  the  next 
section. 

125.  When  the  dividend  is  a  compound  quantity,  the  divi- 
sor may  either  be  placed  under  the  whole  dividend,  as  in  the 
preceding  instances,  or  it  may  be  repeated  under  each  term, 
taken  separately.  There  are  occasions  when  it  will  be  con- 
venient to  exchange  one  of  'these  forms  of  expression  for 
the  other. 

J-j-c         b      c 
Thus  6+c  divided  by  x.  is  either  --  ,  or  —  +  — 

J  XXX 

a+b 
And  a  +  b  divided  by  2,  is  either—  £—  that  is,   half  the 

a      b 
sum  of  a  and  b  ;  or  ^-+  -$  tKat  is,  the   sum  of  half  a  and 

half  6.  For  it  is  evident  that  half  the  sum  of  two  or  more 
quantities,  is  equal  to  the  sum  of  their  halves.  And  the  same 
principle  is  applicable,  to  a  third,  fourth,  fifth,  or  any  other 
portion  of  the  dividend. 


5G  ALGEBRA. 

a — b         a       b 
So  also  a—b  divided  by  2,  is  either— •£-,  or  ~^—~iy 

For  half  the  difference  of  two  quantities,  is  equal  to  the. 
difference  of  their  halves. 

a— 26+/J      a     2b     h             3a  — c       3a        c 
So =——+—.   And = — . 

TO  m     m     m  —x        —  x      —x 

126.  If  some  of  the  letters  in  the  divisor  are  in  each  term 
of  the  dividend,  the  fractional  expression  may  be  rendered 
more  simple,  by  rejecting  equal  factors  from  the  numerator 
and  denominator. 

2am 


Div.     ab 

By       ac 

dhx 
dy 

ahm—3ay 
ab 

ab+bx 
by 

ab     b 

km—3y 

Uuot.      or 
ac      c 

b 

am 
xy 


These  reductions  are  made  upon  the  principle,  that  a 
given  divisor  is  contained  in  a  given  dividend,  just  as  many 
times,  as  double  the  divisor  in  double  the  dividend  ;  triple 
the  divisor  in  triple  the  dividend,  &c.  See  the  reduction  of 
fractions. 

127.  If  the  divisor  is  in  some  of  the  terms  of  the  dividend, 
but  not  in  all  ;  those  which  contain  the  divisor  may  be  divi- 
ded as  in  Art.  116,  and  the  others  set  down  in  the  form  of  a 
fraction. 

ab+d        ab     d  d 

Thus  (ab-t-d)-±a  is  either  ---  ,  or  —  +  —  or  o+—. 

Div.     dxy+rx—hd        2ab-\-ad+x 

By       a?  a  —b 


hd 


y 
Quot.  dy+r-~  -H»+ 


In  the  four  last  articles,  more  than  usual  caution  will  be 
requisite,  in  applying  the  signs.  On  this  subject,  see  the 
next  section. 

128.  The  quotient  of  any  quantity  divided  by  itself  or  iff 
equal,  is  obviously  a 


DIVISION. 

a  Sax  6  a  +  b-3h 

Thus  -  =  l.And        =  l. 


Div.     ax+x       3bd—3d      Aaxy—Aa+8ad 
By      x  3rf  4a 

Quot.  a+1 


Cor.  If  the  dividend  is  greater  than  the  divisor,  the  quoj 
tient  must  be  greater  than  a  unit:  But  if  the  dividend  is  hss 
than  the  divisor,  the  quotient  must  be  less  than  a  unit. 

PROMISCUOUS    EXAMPLES. 

1.  Divide  I2aby+Qabx— IQbbm -\-2Ab,  by  6b. 

2.  Divide  16a  — 12+8^+4- 2,0adx+m,  by  4. 

3.  Divide  (a— 2A)  x(3m->-y)  xx,  by  (a— 2A)  x 

4.  Divide  ahd— 4ac?+3ay — a,  by  Ad— 4(?+3y- 

5.  Divide  ax— ry+ad— Amy— 6-f-a,  by  —  a. 

6.  Divide  amy-\-3my—  mxy+am— d,  by  — 


129.  From  the  nature  of  division  it  is  evident,  that  the 
value  of  the   quotient  depends  both   on  the  divisor  and  the 
dividend.      With  a  given  divisor,  the  greater  the  dividend, 
the  greater  the  quotient.      And  with  a  given  dividend,  the 
greater  the  divisor,  the  less  the  quotient.     In  several  of  the 
succeeding   parts  of    algebra,  particularly   the    subjects  of 
fractions,  ratios,   and  proportion,  it  will  be  important  to  be 
able  to  determine  what  change  will  be  produced  in  the  quo- 
tient, by  increasing  or  diminishing  either  the  divisor  or  the 
dividend. 

If  the  given  dividend  be  24,  and  the  divisor  6 ;  the  quo- 
tient will  be  4.  But  this  same  dividend  may  be  supposed  to 
be  multiplied  or  divided  by  some  other  number,  before  it  is 
divided  by  6.  Or  the  divisor  may  be  multiplied  or  divided 
by  some  other  number,  before  it  is  used  in  dividing  24.  In 
each  of  these  cases,  the  quotient  will  be  altered. 

130.  In  the  first  place,  if  the  given  divisor  is  contained  in 
the  given  dividend  a  certain  number  of  times,  it  is  obvious 
that  the  same  divisor  is  contained, 

In  double  that  dividend,  twice  as  many  times ; 
In  triple  the  dividend,  thrice  as  many  times,  &<% 


58T  ALGEBRA. 

That  is,  if  the  divisor  remains  the  same,  multiplying  the 
ftiriiltiid  by  any  quantity,  is,  in  effect,  multiplying  me  quotient 
by  that  quantity. 

24 
Thus,  if  the  constant  divisor  is  6,  then  -;r=4  the  quotient. 

2x24 
Multiplying  the  dividend  by  2,  ^^ — =2  x  4 

3  x  °4 
Multiplying  the  dividend  by  3,  ———=3x4 


Multiplying  by  any  number  w,  — ;; — =n  x  4. 

131.  Secondly,  if  the  given  divisor  is  contained  in  thcgir- 
en  dividend  a  certain  number  of  times,  the  same  divisor  is 
contained, 

In  half  that  dividend,  half  as  many  times; 

In  one  third  of  the  dividend,  one  third  as  many  times,  kc. 

That  is,  if  the  divisor  remains  the  same,  dividing  the  divi- 
dend by  any  other  quantity,  is,  in  effect,  dividing  the  quotient 
by  that  quantity. 

24 
Thus  7f=4- 

1 24 
Dividing  the  dividend  by  2,  -%—  =  |4. 

Dividing  by  3,  — «~ =^4,  &c. 

132.  Thirdly,  if  the  given  divisor  is  contained  in  the  giv- 
en dividend  a  certain  number  of  times,  then,  in  the  same 
dividend, 

Twice  that  divisor  is  contained  only  half  as  many  times ; 

Three  times  the  divisor  is  contained,  one  third  as  many  times. 

That  is,  if  the  dividend  remains  the  same,  multiplying  the 
divisor  by  any  quantity,  is,  in.  effect,  dividing  the  quotient  by 
that  quantity. 

24 
Thus  ~Q—^ 

24 
Multiplying  the  divisor  by  2, 


24 
Multiplying  by  3,  =  *  &c* 


133.  Lastly,  if  the  giv£n  divisor  is  contained  in  the  givei* 


DIVISION.  59 

dividend  a  certain  number  of  times,  then,  in  the  same  divi- 
dend, 

Half  that  divisor  is  contained,  twice  as  many  times ; 

One  third  of  the  divisor  is  contained  thrice  as  many  times. 

That  is,  if  the  dividend  remains  the  same,  dividing  the  di- 
visor by  any  other  quantity,  is,  in  effect,  multiplying  the  quo- 
tient by  that  quantity. 

24 
Thus  -g-=4, 

24 

Dividing  the  divisor  by  2,  —  =2  X  4 

^b 
24 
Dividing  by  3,  —=3x4 


For  the  method  of  performing  division,  when  the  divisor 
and  dividend  are  loth  compound  yuaalitie^  sec  one  of  the 
•following  sections. 


y 

, 


• 

. 

SECTION    VI. 


FRACTIONS.* 

134  T^XPRESSIONS  in  the  form  of  fractions  o.-- 
cur  more  frequently  in  algebra  tluui  in  arith- 
metic. Most  instances  in  division  belong  to  this  class.  In- 
deed the  numerator  of  every  fraction  may  be  considered  as 
a  dividend,  of  which  the  denominator  is  a  divisor. 

According  to  the  common  definition  in  arithmetic,  the 
denominator  shows  into  what  parts  an  integral  unit  is  sup- 
posed to  be  divided;  and  the  numerator  shows  how  many  of 
these  parts  belong  to  the  fraction.  I3ut  it  makes  no  differ- 
ence, whether  the  whole  of  the  numerator  is  divided  by  the 
denominator;  or  only  one  of  the  integral  units  is  divided, 
and  then  the  quotient  taken  as  many  times,  as  the  number  of 
units  in  the  numerator.  Thus  |  is  the  same  as  ? +  -j  +  -j.  A 
fourth  part  of  three  dollars,  is  equal  to  three  fourths  of  one 
dollar. 

135.  The  value  of  a  fraction,  is  the  quotient  of  the  nume- 
rator divided  by  the  denominator. 

6  ab 

Th"s  the  value  of  ~r  is  3.      The  value  of  -r  is  a. 
lf  2  b 

From  this  it  is  evident,  that  whatever  changes  are  made  m 

the   terms  of  a  fraction ;  if  the  quotient  is  not  altered,  the 

value  remains  the  same.    For  any  fraction  therefore,  we  may 

substitute  any  other  fraction  which  will  give  the  same  quotient. 

4      10     &ba      Sdrx     6  +  2 

Thus  -^==,—=—j — =TI — =^ry  &ic.     For  the  quotient 

in  each  of  these  instances  is  2. 

136.  As  the  value  of  a  fraction  is  the  quotient  of  the  nu- 
merator divided  by  the  denominator,  it  is  evident,  from  Art. 
128,  that  when  the  numerator  is  equal  to  the  denominator, 
the  value  of  the  fraction  is  a  unit;  when  the  numerator  is 

*  Horsley's  Mathematics,   Camus'  Arithmetic,  Emerson.   Euler, 
Saunderson,  and  Ludlam. 


FRACTIONS.  61 

less  thin  the  denominator,  the  Value  is  lete  than  a  unit ;  and 
when  the  numerator  is  greater  than  the  denominator,  the  val- 
ue is  greater  than  a  unit. 

The  calculations  in  fraction's  depend  on  a  few  general 
principles,  which  will  here  be  stated  in  connection  with  each 
•ther. 

137.  If  the  denominator  of  a,  fraction  remniiif  tie  same, 
multiplying  the  NUMERATOR  by  any  quantity,  is  multiplying  the 


the  quotient.  And  by  Art.  130  and  131,  multiplying  the  div- 
idend is  in  effect  multiplying  the  quotient,  and  dividing  the 
dividend  is  dividing  the  quotient. 

ab     Sab      laid 

.Thus,  in  the  fractions  —  ,    ~~~,    ~~, 
a'      a   '       a    ' 

The  quot's  or  values  are  b,      35,      lid, 

Here  it  will  be  seen  that,  while  the  denominator  is  not  al- 
tered, the  value  of  the  fraction  is  multiplied  or  divided  by 
the  same  quantity  as  the  numerator. 

Cor.  With  a  given  denominator,  the  greater  the  numera- 
tor, the  greater  will  be  the  value  of  the  fraction  ;  and,  on  the 
other  hand,  the  greater  the  value,  the  greater  the  nume- 
rator. 

1  38.  If  the  numerator  remains  the  same,  multiplying  the  de- 
nominator by  any  quantity,  is  dividing  the  value  by  that  quan- 
tity; and  dividing  the  denominator,  is  multiplying  the  value. 
For  multiplying  the  divisor  is  dividing  the  quotient  ;  and 
dividing  the  divisor  is  multiplying  the  quotient.  (Art.  132, 
133.) 


In  the  fractions-^-,  -^  -£-,  -r-,  &c. 

The  values  are       4«,       2a,       Sa,    '  24ffl,  &.c. 

Cor.  With  a  given  numerator,  the  greater  the  denomina-» 
tor.  the  hss  will  be  the  value  of  the  fraction  ;  and  the  less 
fee  value,  the  greater  the  denominator 

j  o9.  From  the  two  last  articles  it  follows,  that  dividing  the 
numerator  by  any  quantity,  will  have  the  same  effect  on  the 
value  of  the  fraction,  as  multiplying  the  denominator  by  that 
quantity  ;  and  multiplying  the  numerator  will  have  the  same 
effect,  as  dividing  the  denominator. 

140.  It  is  also  evident,  from  the  preceding  articles,  that 
if  the  numerator  and  denominator  be  both  multiplied,  or  both 


82  ALGEBRA. 

divided,  by  the   samp,  quantity,  the  value  of  the  fraction  will 
not  be  altered. 

bx      abx      3bx      ^bx     $abx 

Thus  T==~a6"="3r==^r==^6  **•      For  m  cach  of 
these  instances  the  quotient  is  x. 

141.  Any  integral  quantity  may,  without  altering  its  value, 
be  thrown  into  the  form   of  a  fraction,  by  multiplying  the 

3uantity  into  the  proposed  denominator,  and  taking  the  pro- 
uct  for  a  numerator. 

a     ab     ad-\-ah     6adh 

Thusa=T=y=-^+F=-^r,kc.  For  the  1uotient 
«f  each  of  these  is  <?. 

dx+hx  2drr+2dr 

So  d+h~— — •         Andr+l*— ^jr— . 

142.  There  is  nothing  perhaps,  in  the  calculation  of  alge- 
braic fractions,  which  occasions  more  perplexity  to  a  leanv 
er,  than  the  positive  and  negative  signs.     The  changes  in 
these  are  so  frequent,  that  it  is  necessary  to  become  familiar 
with  the  principles  on  which  they  are  made.      The  use  of 
the  sign   which  is  prefixed  to  the  dividing  line,  is  to  show 
whether  the  value  of  the  whole  fraction  is  to  be  added  to,  or 
subtracted  from,  the   other  quantities  with  which  it  is  con- 
nected. (Art.  43.)     This  sign,  therefore,  has  an  influence  on 
the  several  terms  taken  collectively.      But  in  the  numerator 
and  denominator,  each  sign  affects  only  the  single  term  to 
which  it  is  applied. 

ab 
The  value  of  -r  is  a.  (Art.  135.)     But  this  will  become 

negative,  if  the  sign  —  be  prefixed  to  the  fraction. 

ab  ab 

Thus  y-r--r=y+«.     Bat  y—-r=y  —  a. 

So  that  changing  the  sign  which  is  before  the  whole  frac- 
tion, has  the  effect  of  changing  the  value  from  positive  to 
negative,  or  from  negative  to  positive. 

Next,  suppose  the  sign  or  signs  of  the  numerator  to  b« 
changed. 

ab  —ab 

By  Art.  123,  -^  —  +a.          But  -^-  =— a. 

ab—bc                                      —ab+bc 
And — -7 — —+a—c.  But 7 =—  0+e. 

That  is,  by  changing  all  the  signs  of  the  numerator,  the 

• 


FRACTIONS.  63 

ralue  of  the  fraction  is  changed  from  positive  to  negative,  or 
the  contrary. 

Again,  suppose  the  sign  of  the  denominator  to  be  chan- 
ged. 

ab  ab 

As  before -5-= -f- a.  But  — r=—  «• ' 

.  '          T. 

143.  We  have,  then,  this  general  proposition;   If  the  sign 

prefixed  to  a  fraction,  or  all  the  signs  of  the  numerator,  or  all 
the  signs  of  the  denominator  be  changed;  the  value  of  the  frac- 
tion will  be  changed,  from  positive  to  negative,  or  from  nega- 
tive to  positive.' 

From  this  is  derived  another  important  principle.  As 
each  of  the  changes  mentioned  here  is  from  positive  to  neg- 
ative, or  the  contrary ;  if  any  two  of  them  be  made  at  the 
same  time,  they  will  balance  each  other. 

Thus,  by  changing  the  sign  of  the  numerator, 
ab  —ab 

-r-=-t-a  becomes  — 7—  =  —  a. 

But,  by  changing  both  the  numerator  and  denominator,  h 

—ab 
becomes  ~^jT=  +  o,  where  the  positive  value  is  restored. 

By  changing  the  sign  before  the  fra.ction> 
ab  ab 

w+-7-=w-f  a  becomes  y^--r=y—a. 
y     b     y  y     i)     y 

But,  by  changing  the  sign  of  the  numerator  also,  it  be** 

— ab 
comes  y — — v—  where  the  quotient — a  is  to  be  subtracted  from 

y,  or  which  is  the  same  thing,  (Art.  81,)  -fa  is  to  be  added, 
making  y+a  as  at  first.  Hence, 

144.  If  all  the  signs  both  of  the  numerator  and  denomina- 
tor, or  the  signs  of  one  of  these  with  the  sign  prefixed  to  th& 
whole  fraction,  be  changed  at  th&  same  time,  the  value  of  the- 
fraction  will  not  be  altered. 

6      -6          -6  6 

Thus  —=— ^-  = -——=-— -^  = +  3. 

_6 --6__       6__      _^6__ 

Hence  the  quotient  in  division  may  be  set  down  in  differ- 

a       —c        a       c 
ent  ways.      Thus  (a— c)-±b,  is  either  ~T+~jT~>  orT~~~~b' 

The  latter  method  is  the  most  common.  See  the  exaiu- 
elt*;  in  Art.  127. 


64  ALGEBRA. 


REDUCTION  op  FRACTIONS. 

145.  From  the  principles  which  have  been  stated,  are  de- 
rived the  rules  for  the  Reduction  of  fractions,  which  are  sub- 
stantially the  same  in  algebra,  as  in  arithmetic. 

Jl  fraction  may  be  reduced  to  lower  terms,  by  dividing  both 
the  numerator  and  denominator,  by  any  quantity  which  will  di- 
vide them  without  a  remainder.  According  to  Art.  140,  this 
will  not  alter  the  value  of  the  fraction. 

ab     a  6dm      3m  7m       1 

Thus  -r= — .     Andin~  =  ~l — .      And -r~  = — . 
cb      c  Sdy      Ay  Imr      r 

In  the  last  example,  bath  parts  of  the  fraction  are  divided 
by  the  numerator.  The  reduced  numerator  must  therefore 
'be  a  unit.  (Art.  128.) 

fl+ic  I  am  +  ay      a  [121.) 

Again  j — ~j~\ = —  (Art.  118.)  And  7—  77- =-7.   (Art. 

(a+bc)xm       m  ^  '  bm  +  by      b 

If  a  letter  is  in  every  term  both  of  the  numerator  and  de- 
nominator, it  may  be   cancelled,  for  this  is  dividing  by  that 
letter.  (Art.  120.) 
rrn,      3g»H-ay_ 

ad+ah  =     d+h  y-y^ 

146.  Fractions  of  different  denominators  may  be  reduced  to 
a  common  denominator,  by  multiplying  each  numerator  into  all 
the  denominators  except  its  oivn,  for  a  new  numerator  ;  and  all 
the  denominators  together,  for  a  common  denominator. 

a  c  m 

Ex.  1.   Reduce  -j~,   and   --y,   and   — to  a  common  de- 
o  a  y 

nominator. 

axdxy=ady   Y 

cxbxy=cby     >    the  three  denominator?. 

mxbxd=mbd  ) 

bxdxy=bdy      the  common  denominator. 

ady  bey  bdm 

The  fractions  reduced  are  -7-7-,   and  T-J-,  and  "Or- 

Here  it  will  be  seen,  that  the  reduction  consists  in  multi- 
plying the  numerator  and  denominator  of  each  fraction,  into 
all  the  other  denominators.  This  does  not  alter  the  value. 
(Art.  140.) 

dr  2Jt  6c 

2.  Reduce  -^-,   and   --,   and  — . 

dgry  Ghmy  1  Scgrn 

Ans.     -77- — ,   and   TT ,    and  ~^~ 

' 


FRACTIONS. 

2  a  r+1 

3.  Reduce  -g-»    and  —  »    and      v 


-_ 
w''    and  '    and  ' 


1  1 

4.  Reduce  -  -.-r»    and  ---  F* 
«+6  a—  b 

a—b  a+b 

fins.  -  TV'    and  -    —  TT* 
aa—bo  aa—ob 

. 

After  the  fractions  have  been  reduced  to  a  common   de- 
nominator, they  may  be  reduced  to  lower  terms,  by  the  rule  v 
in  the  last  article,  if  there  is  any  quantity,  which  will  divide 
the  denominator,  and  all  the  numerators,  without  a  remain- 
der. 

An  integer  and  a  fraction  are  easily  reduced  to  a  common 
denominator.  [(Art.  141.) 

b  a  b          ac          b 

Thus  a  and  —  are  equal  to  -7-  and  —  »  or  —  and  — 
c  1  c  c          c 

. 
h    d  amy  bmy       hy      dm 

And  a,  b,  —  >  —are  equal  to  --  >  --  >    --  f  - 

'my  my     my       my     my 

. 

147.  To  reduce  an  improper  fraction  to  a'mixed  quantity, 
divide  the  numerator  by  the  denominator,  as  in  Art.  127. 

ab  +  bm+d 
Thus  -    —  --- 


dx+d — 7A+y  7A      y 

And  ~j  =a?-j- 1  — ~T-}-~T* 

a>  a       a 

am— a -\-ady— hr 

Reduce -" •>     to  a  mixed  quantity, 

a 

For  Jfie  reduction  of  a  mixed  quantity  to  an  improper 
fractkfn,  see  Art.  150.  And  for  the  reduction  of  a  compound 
fraction  to  a  simple  one,  see  Art.  160. 


ALGEBRA 


ADDITION  or  FRACTIONS. 

148.  In  adding  fractions,  we  may  either  write  them  one 
after  the  other,  with  their  signs,  as  in  the  addition  of  inte- 
gers, or  we  may  incorporate  them  into  a  single  fraction,  by 
the  following  rule : 

Reduce  the  fraction*  to  a  common  denominator,  make  the 
signs  before  them  all  positive,  and  then  add  their  numerators. 

The  common  denominator  shows  into  what  parts  the  inte- 
gral unit  is  supposed  to  be  divided ;  and  the  numerators  show 
the  number  of  these  parts  belonging  to  each  of  the  fractions.. 
(Art.  134.)  Therefore  the  numerators  taken  together  show 
the  whole  number  of  parts  in  all  the  fractions. 

211  3111 

Thus  — =-^-4--y  And  ^  =— +— -f  — * 

2  -.  3       1       1       1115 

Therefore  -^+~^=~^+'y+~^"~r-~7~4"~7~=~7r" 

The  numerators  are  added,  according  to  the  rules  for  th«r 
addition  of  integers.  (Art  69,  &c.)  It  is  obvious  that  the  sum* 
is  to  be  placed  over  the  common  denominator.  To  avoid 
the  perplexity  which  might  be  occasioned  by  the  signs,  it 
will  be  expedient  to  make  those  prefixed  to  the  fractions- 
uniformly  positive.  But  in  doing  this,  care  must  be  taken 
not  to  alter  the  value.  This  will  be  preserved,  if  all  the 
signs  in  the  numerator,  are  changed  at  the  same  time  with 
that  before  the  fraction.  (Art.  144.) 

•  -\  r 

2,          4  2+4         6 

Ex.  1.  Add  TQ  and  rjg  of  a  pound.     Ans.  •••  fi  -  or  Tg* 

• 

It  is  as  evident  that  T\,  and  T\-  of  a  pound,  are  -fs  of  & 
pound,  as  that  2  ounces,  and  4  ounces,  are  6  ounces. 

a          c 
2.  Add  -T-  and  -v    First  reduce  them  to  a  common  denomi- 

ad        be  ad-\-bc 

»ator.  They  vi  ill  then  be  7-7  and  o'    and  their  sum  — T-» — 


FRACTIONS. 

2r+d 


3.  Given  -7  and  —  ""ST""'   to  find  their  sum.    / 


m 
Ans.-jand—  ^-= 

a  b—  m     a      —  J-fm     cy—bd+dm 

A.  ~ 


d        —am        dy       ^^J^ffy       f^Tl^.  (Art. 
=~~=    ~~~        ~~ 


a  b        aa—ab  +  ab+bb     aa  +  bb 

£.   :~~  T  and  -  r=  —  ;  —  i  -  1  —  n—~  —  TT*   (Art.  77.) 
a+b         a—  b     aa+ab—ab  —  bb     aa—bb 

—a        —h  -4      -16 

7.    Add  —    to  -—  •         8.  Add  -—  -to—  -    Am.  -6. 


149.  For  many  purposes,  it  is  sufficient  to  add  fractions 
in  the  same  manner  as  integers  are  added,  by  writing  them 
.one  after  another  with  their  signs.  (Art.  69.) 

a          3  d  a      3       d 

Thus  the  sum  of  y  and  y  and  — ^->    is  j  +  -— ^ 

In  the  same  manner,  fractions  and  integers  may  be  ad- 


d  h  d      h 

The  sum  of  «  and  —  and  3»i  and  —  —  .  is  &-{-3m+  —  —  —  • 
y  r,  y      r 

150.  Or  the  integer  may  be  incorporated  with  the  fraction, 
1)y  giving  to  the  former  the  denominator  of  the  latter,  and 
then  adding  the  numerators.  See  Art.  141. 

b          a      b     am     b      am-}-b  • 
The  sum  of  a  and  -,   is  T  +  -=-  +"-—  - 


The  sum  of  3J  and  -   —  j  -is 


m—  y  m—y 

Incorporating  an  integer  with  a  fraction,  is  the  same  as 
reducing  a  mixed  quantity  to  an  improper  fraction.  For  a 
mixed  quantity  is  an  integer  and  a  fraction.  In  arithmetic, 
these  are  generally  placed  together,  without  any  sign  be* 


tid  ALGEBRA. 

tvveen  them.     But  in  algrbra,  they  are  distinct  terms.    Thus 
2-^  is  2  mud  -J,  which  is  the  same  as  2+-j. 


Ex.  1  .  Reduce  a  -f-  ~/  to  an  improper  fraction.  Ans.  —  r~~ 

"3  3  3     24+3     27 

2.  Reduce  6  T   Aas.  6  ^=C4-  £  =-j-=^- 

r  Tim—  dm  +  dh  —  dd—  r 

3.  Reducje  m+d—  T~y     Ans.  -  ^—  ^  — 

d  b+d  h 

4.  Reduce  1+7"    Ans.  —7  —        5.  Reduce  I——* 


c  - 

6.  Reduce  6+-T  --  7.  Reduce  34-  ~  iT 

a  —  w  «>« 


SUBTRACTION  OF  FRACTIONS.' 

151.  The  methods  of  performing  subtraction  in  algebra, 
depend  on  the  principle,  that  adding  a  negative  quantity  is 
equivalent  to  subtracting  a  positive  one;  and  v.  v.  (An.  81.) 
For  the  subtraction  of  fractions,  then,  we  have  the  following 
simple  rule.  Change  the  fraction  to  be  subtracted,  from  posi- 
tive to  negative,  or  the  contrary,  and  then  proceed  as  in  addi-> 
tion.  (Art.  148.)  In  making  the  required  change,  it  will 
be  expedient  to  ,alter,  in  some  instances  the  signs  of  the  nu- 
merator, and  in  others,  the  sign  before  the  dividing  line, 
(Art.  143,)  so  as  to  leave  the  latter  always  affirmative. 

a  h 

Ex.  1.    From  7  >    subtract    "rr 
u  m 

,         •  h  —h 

First  change  — >    the  fraction  to  be  subtracted,  to 

0     m  f        m 

Secondly,  reduce  the  two  fractions  to  a  common  denomi- 

am  —bh 

rmtor,  making  -7;-  and  -7-- 

Thirdly,  take  the  sum  of  the  numerators,   am—bh  : 
This,   placed  over  the  common  denominator,  gives  the 

am — bh 
answer  required,  ~~biii~1' 


FRACTIONS.  69 


h  <<d-\-<1y—kr 

»    suract 


- 
2.   From  --  »    subtract    -7-        Ans.  —  —  -7 


a  d—b  ay—dm  +  lm 

3.    From  —  subtract  —  —  •    Ans.  -  ~~— 


—  9« 

4.  From  —  ;,  —  >    subtract  —  5—    Ans.  —7^  —  • 
4  o  12 

6—  d  b  b—d     b      by—dy+bm 

5.  From  —  subtract  --•.  Ans.-^+y^      —  -  . 

-    a+1  rf-1  3  4 

6.  From  —  7—  subtract  ----       7:  From  —  subtract  -r- 

a  m  a  •  b 

152.  Fi-actions  may  also  be  subtracted,  like  integers,  by 
setting  them  down,  after  their  signs  are  changed,  without 
reducing  them  to  a  common  denominator. 

t  ft     .  d  a     .  d 

Ex.  1.  From  -r  subtract  -r-' 


h  ,  h-\-d  h 

2.  From  -  subtract  -  --•       Ans.  -+—  • 

In  tlie  same  manner,  an  integer  may  be  subtracted  from  a 
fraction,  or  a  fraction  from  an  integer. 

b  b 

3.  From  a  subtract  —  •  Ans.  «—  —  • 

m  m 

153.  Or  the  integer  may  be  incorporated  with  the  frac- 
tion, as  in  Art.  1  50. 

h  h  Ti—my 

Ex.   1.  From  —  subtracts.    Ans.  —  —  m=  —    — 

'  y  y          y 

b  h 

2.  -From  4«+  —  subtract  3a  —  -y 

C  c* 

bd+hc     acd  -\-bd-\-  he 


70  ALGEBRA. 

&—  t  c-b 

3.  From  1+--  subtract  —j—   Ans. 


4.  From  a-J-3A—  —  —  subtract  Sa—  ft-f  —  ~—  - 


MULTIPLICATION  OP  FRACTIONS. 

154.  By  the  definition  of  multiplication,  multiplying  by  » 
fraction  is  taking  a  part  of  the  multiplicand,  as  many  times, 
as  there  are  like  parts  of  an  unit  in  the  multiplier.  (Art.  90.) 
Now  the  denominator  of  a  fraction  shows  into  what  part» 
the  integral  unit  is  supposed  to  be  divided ;  and  the  numera- 
tor  shows  how  many  of  those  parts  belong  to  the  giv^n  frac- 
tion. In  multiplying  by  a  fraction,  therefore,  the  multipli- 
cand is  to  be  divided  into  such  parts,  as  are  denoted  by  the 
denominator ;  and  then  one  of  these  parts  is  to  be  repeated, 
as  many  times,  as  is  required  by  the  numerator. 

3 

Suppose  a  is  to  be  multiplied  by  -r- 

A  fourth  part  of  a  is  -r- 


j 

This  taken  3  times  is  "4  +"4  +  4"=T'  (Art' 148'^ 

«  3 

Again,  suppose  ~T  is  to  be  multiplied  by  ~JT 

One  fourth  of  y  is  -rg    (Art.  138.) 

er      a      «      30 

This  taken  3  times  is  46"^46^"4A:=:4A 

the  product  required. 

In  a  similar  manner,  any  fractional  multiplicand  may  be 
divided  into  parts,  by  multiplying  the  denominator;  and  one 
of  the  parts  may  be  repeated,  by  multiplying  the  numerator. 
We  have  then  the  following  rule  : 


FRACTIONS.  71 

155.  To  multiply  fractions,  multiply  the  numerators  togeth- 
er, for  a  new  numerator,  and  the  denominators  together,  for 
*  new  denominator. 


Ex.  1.  Multiply  —  into  —   Product   -—  - 


a+d  Ah  Aah+Adh 

2.  Multiply  --mto--   Product          _      ' 


.  4 

3.  Mult.  -£=  ~3n)-     99) 


4.  Mult.  -  -  --  into  -7-7-7  •  Prod.       ,    ,  — 

y  d+1  dy+y 

a+h         A—m,  1  3 

5.  Mult.     -     into—--    6.  Mult.  —  -^  into  T 


156.  The  method  of  multiplying  is  the  same,  when  there 
are  more  than  two  fractions  to  be  multiplied  together. 


Ill/ 

1.  Multiply  together  y  -g>  and  —  •    Product  TT~- 

a      c  .  ac  m  .  «cm 

For  TXT  1S>  oy  the  last  article  r>  and  this  into  — ,isr^ — 
b      a  odr  y '    oay 

2a  h-d   b  1  2abh—2abd 

2.  Multiply  — y 5  — '  and r-  Product 

*  J   m      y       c  r—l  cmry—cmy 

3+6    Id  ad  «— 6          3 

3.  Mult. »-r»  and  — rz>*    4.  Mult.  T~»  TTT,  and-=- 

n       A  r+2  Ay  a+1'         7 

157.  The  multiplication  may  sometimes  be  shortened,  by 
rejecting  equal  factors,  from  the  numerators  and  denom- 
inators» 

a  .         h  d  dh 

I.  Multiply  —  into  —  and —  Product  — - 
1  v    r  a  y  ry 

Hers  a  being  in  one  of  the  numerators,  and  in  one  of  the 


.S2  .  ALGKm;\ 

dcnorniniitors,  may  be  omitted.      If  it  be  retained,  the  pro- 

adh 
duct  will  be  -~~~'    But  this  reduced  to  lower  terms,  .by 

dh 

Art.  145,  will  become  -  -  as  before. 

.  ry 

ad          m          ah  ah 

2.  Midt.  —  into  TT-  and  <rr    Product  ~^- 

in  on  2ti  b 

irtu  It  is  necessary  that  the  factors  rejected  from  the  numera- 
tors be  exactly  equal  to  those  which  are  rejected  from  the 
Denominators.  Jn  the  last  example,  a  being  in  two  of  the 
numerators,  and  in  only  one  of  the  denominators,  must  be 
retained  in  one  of  the  numerators. 

7  , 

a  +  d         my  am  +  dm 

3.  Mult. into  — f-     Prod.  -  -7 

y  an.  an 

Here,  though  the  same  letter  a  is  in  one  of  the  numera- 
tors, and  in  one  o-f  the  denominators,  yet  as  it  is  not  in  every 
term  of  the  numerator,  it  must  not  be  cancelled. 


am  -4-  d     *""' .    h 


>* 
3r    .  3,77 


4.  MulU — 7 — ;into  -r  and  -=—  Prod. 

h  m  5a  5am 

If  any  difficulty  is  found,  in  making  these  contractions,  it 
ill  be  better  to  perform  the  multiplication,  without  omitting 
any  of  the  factors;  and  to  reduce  the  product  to  lower- 
terms  afterwards. 

158.  When  a  fraction  and  an  integer  are  multiplied  togeth- 
er, the  numerator  of  the  fraction  is  multiplied  into  the  inte- 
ger. The  denominator  is  not  altered;  except  in  casr^ 
where  division  of  the  denominator  is  substituted  for  multi- 
plication of  the  numerator,  according  to  Art.  139. 

• 

m     am  a  am      am 

Thu3axy  =  y  For;a=T;  and  jxy  =  y 

x     h+l      hrx+rx  I       a 

So  rx-^-x— 3~ = — ^ And0Xy=-£"     Hence? 

_  ^i 

. 


FRACTIONS.  7S 

159.  JL  fraction  is  multiplied  into  a  quantity  equal  to  its  dv 
nominator,  by  cancelling  the  denominator. 

a  a  ab 

Thus  -f-  x  b  =•«.  For  -7-  x  b =~r*  But  the  letter  I,  be- 
ing in  bofh  the  numerator  and  denominator,  may  be  set 
aside.  (Art.  145.) 

So  -^—  x(a-^)=3m.    And  -3  — —  x(3+m)^+3(f. 

On  the  same  principle,  a  fraction  is  multiplied  into  any 
factor  in  its  denominator,  by  cancelling  that  factor. 

a  ay      a  k  h 

^  -by^by^V   And24x6==T 

160.  From  the  definition  of  multiplication  by  a  fraction, 
it  .follows  that  what  is  commonly  called  a  compound  fraction  ,* 

a, 
is  the  product  of  two  or  more  fractions.      Thus  |  of  ~r    is 

a  a  a 

|  X  y    For  |  of  -£-»    is  |  of  y  taken  three  times,  that  is, 

a      a      a  a 

TA  +  AA+AA'    But  this  is  the  same  as  -y-    multiplied   by  |. 

(Art.  154.) 

Hence,  reducing  a  compound  fraction  to  a  simple  one,  is  the 
same,  as  multiplying  fractions  into  each  other. 

2     ,    a  2a 

Ex.   1.   Reduce  Trof7-;~7;>   Ans. 


7UI6+2          "  76+14 


2.    Reduce  -^  of -g  of 0^'  '!«  Ans.  Qnl  '  ^—< 


3.  Reduce  -^of-^ofo  —  V  Ans. 


161.  The  expressions  -|a,  |&,  4y>  &£C"  are  equivalent  fp 
2a      6     4y  x  2 

T'    T'    ^7J         ^or  ^a  *s  ^  °^  a'  w^ic^  is  equal  to  -&  x  a=» 

2«  6 

-3-  (Art.  158.)    So|6=|x5=y 

*  By  a  compound  fraction  is  meant  a  fraction  of  a  fraction,  and  not 
91  fraction  whose  numerator  or  denominator  is  a  compound  quantity. 

K 


ALGEBRA. 


DIVISION  OF  FRACTIONS. 

• 

-     162.  To  divide  one  fr  action  ly  another,  invert  the  divisor, 
and  then  proceed  as  in  multiplication.    (Art  155.) 

a          c  a      d     ad 

Ex.  1.  Divide-^  by  ^   Ans.  JX~=^' 

To  understand  the  reason  of  the  rule,  let  it  be  premised, 
the  product  of  any  fraction  into  the  same  fraction  in- 
rerted  is  always  a  unit. 


a      b      ab  d,       A+'V     dh4-dy          (  AH 

Thus  Tx-=-7  =  l.     Andrr-X—  T^=^TT/  =  1.  ) 

b      a     ab  h+y       d       dh+dy          128.) 

But  a  quantity  is  not  altered  by  multiplying  it  by  a  unit. 
Therefore  if  a  dividend  be  multiplied,  first  into  the  divisor 
inverted,  and  then  into  the  divisor  itself,  the  last  product 
will  be  equal  to  the  dividend.  Now,  by  the  definition,  art. 
115,  "  division  is  finding  a  quotient,  which  multiplied  into 
the  divisor  will  produce  the  dividend."  And  as  the  dividend 
multiplied  into  the  divisor  inverted  is  such  a  quantity,  the 
quotient  is  truly  found  by  the  rule. 

This  explanation  will  probably  be  best  understood,  by  at- 
tending to  the  examples.  In  several  which  follow,  the  proof 
of  the  division  will  be  given,  by  multiplying  the  quotient  into 
the  divisor.  This  will  present,  at  one  view,  the  dividend 
multiplied  into  the  inverted  divisor,  and  into  the  divisor 
itself. 

m        3h  m      y      my 

2.  Divide        by  Ans. 


my      3  h     3>hmy      m 

Proof.  ;s~jrX —  =  ~^~fT=^j  the  dividend. 
6dh       y      Qdny     2a 

x+d       5d  x+d     y     xy+dy 

'3.  Divide  -     -by —     Ans. -x^=— -TJ—  • 

r       J    y  r        ad         ouf 


&d     5dxy -\-5ddy     x-\-d 

Proof.  — ^j — X — = FT —   — =— 

oar        y  5dry  r 

&dh         4fhr  4-dh        a       ad 

4.  Divide  —  by  — •      Ans.  —  X-^T=-(Art.l 

ad     4-hr     Aadhr      &dh 
Proof:  -x--=-—=T- the  dividend. 


FRACTIONS. 

36J        187*  md      lOy     Ady 

5.  Divide  -y-  by  Ans.  -5-  X  ^  =-A 


ISA 
Proof  --x     ----,  the  dividend. 


aft-1  h—m 

6.  Divide  —  ;  ~  —  by  -  7.  Divide  —  , 

3y       '      a?  4 

163.  When  a  fraction  is  divided  by  an  integer,  the  de- 
nominator of  the  fraction  is  multiplied  into  the  integer. 

a  a 

Thus  the  quotient  of  y  divided  by  T»,  is  T~  • 

m  a     m      a      1       a 

For  m—  y  ;  and  by  -the  last  article,  y-j-  J—  £~  X  .«  =^ 

i  11        i  3       •  _3_    j: 

So          i'^rh=-        7XT~=~r     TT*    And  ~r  -7-  6  =777=  "75"* 
a—b  a—  oh      ah—  oh  4  24     8 

In  fractions,  multiplication  is  made  to  perform  the  office 
of  division  ;  because  division  in  the  usual  form  often  leaves 
a  troublesome  remainder.  But  there  is  no  remainder  in 
multiplication.  In  many  cases,  ftiere  are  methods  of  short- 
ening the  operation.  But  these  will  be  suggested  by  prac- 
tice, without  the  aid  of  particular  rules. 

164.  By  the  definition,  art.  49,  "the  reciprocal  of  a  quan- 
tity, is  the  quotient  arising  from  dividing  a  unit  by  that 
quantity." 

a  abb 

Therefore,  the  reciprocal  of  -r.  is  1  —  x=l  X~~=  —  That  is. 

o  '  b  a      a 

The  reciprocal  of  a  fraction.  is  the  fraction  inverted. 

b     .   m+y  1 

Thus  the  reciprocal  of      .     is  —  7—  ;  the   reciprocal  of  IT 

3y 
is  -r-  or  3y;  the  reciprocal  of  \  is  4.     Hence   the   recipro- 

cal of  a  fraction  whose  numerator  is  1,  is  the  denominator 
.of  the  fraction. 

Thus  the  reciprocal  of  —  is  a:  of  —  .—  r,  is  a-\-b.  &e. 
a  a  +  oj 


165.  A  fraction  sometimes  occurs  in  the  numerator  or  df- 

1« 
iiominator  of  another  fraction,  as  ~r"   It  is  often  . j>nvement, 


7«  ALGEBRA. 

in  the  course  of  a  calculation,  to  transfer  such  a  fraction!, 
from  the  numerator  to  the  denominator  of  the  principal 
fraction,  or  the  contrary.  That  this  may  be  done,  without 
.altering  the  value,  if  the  fraction  transferred  be  inverted^  is 
evident,  from  the  following  principles  : 

First,  Dividing  by  a  fraction,  is  the  same  as  multiplying  by 
the  fraction  inverted.  (Art.  162.) 

Secondly,  Dividing  the  numerator  of  a  fraction  has  the 
same  effect  on  the  value,  as  multiplying  the  denominator;  and 
multiplying  the  numerator  has  the  same  effect,  us  dividing 
the  denominator.  (Art.  139.) 

•    -yfl  a 

Thus  in  the  expression  -—,  the  numerator  of  -•  is  multi- 

'X  vLf 

plied  into  |.  But  the  value  will  be  the  same,  if,  instead 
of  multiplying  the  numerator,  we  divide  the  denomi- 
nator by  |,  that  is,  multiply  the  denominator  by  -j. 

\a      a  h       yA 

Therefore  •—  =—  •  So  -7—  =—  • 


= 


m 
a—x      -a— 


166.  Multiplying  the  numerator  is  in  effect  multiplying 
the  value  of  the  fraction.  (Art.  137.J  On  this  principle,  a 
fraction  may  be  cleared  of  ft  fractional  co-efficient  whicjj 
occurs  in  its  numerator. 

la     3      a     3«  -la      1      <t      a 

Thus  JT-=-^XX=^!'     And—  =-vX—  =r~* 
6      56     5o  9  5 


1     h+x    h+x  |.r     3x 

And     ~  X~~~-~-" 


3a     3      a     |a 
On  the  other  hand,  —  =—  x—  =  —  • 


a      A 


167.  But  multiplying  the  denominator,  by  another  fraction, 
is  in  effect  •dividing  the  value;  (Art.  138.)  that  is,  it  is  multi- 
plying the  value  by  the  fraction  inverted.  The  principal 
fraction  may  therefore  be  cleared  of  a  fractional  co-efi'tcicnt, 
which  occu-s  in  its  denominator* 


FRACTIONS. 


77 


a      a      3      a      5      5a  a      7a 

^^     ^     X='     And       =' 


a+h 


21/t 


7a      a 
On  the  other  hand,  r-=j- 


**       ^&  ^ 


--&—•=_  ^L  &  c       **rt 

Jr~*    ^  T£" 


j 

-~ 


SECTION  VM. 


SIMPLE  EQUATIONS. 


\RT    168    T^^  subjects  of  the  preceding  sections  are 
•   introductory  to  what  may  be  considered  the 
peculiar  province  of  algebra,  the  investigation  of  the  values 
of  unknown  quantities,  by  means  of  equations. 

Jin  equation  is  a  proposition,  expressing  in  algebraic  charac- 
ters, the  equality  between  one  quantity  or  set  of  quantities  and 
another.  Thus  z4-a=£>  +  c,  is  an  equation,  in  which  the 
sum  of  x  and  a,  is  equal  to  the  sum  of  b  and  c.  The  quan- 
tities on  the  two  sides  of  the  sign  of  equality,  are  sometimes 
called  the  members  of  the  equation ;  the  several  terms  on 
the  left  constituting  the  first  member,  and  those  on  the  right, 
the  second  member.  In  the  equation  a+y=d— x,  the  first 
member  is  cr+y,  and  the  second  d—x. 

169.  The  object  aimed  at,  in  whit  is  called  the  resolution 
or  reduction  of  an  equation  is  lofoid  the  value  of  the  unknown 
quantity.     In  tke  first  statement  of  the  conditions  of  a  prob- 
lem,  thfe   known   and   unknown    quantities   are   frequently 
thrown  promiscuously  together.      To  find  the  value  of  that 
which  is  required,  it  is  necessary  to  bring  it  to  stand  by  it- 
self, while  all  the  others  are  on  the  opposite  side  of  the 
equation.      But,  in  doing  this,  care  'must  be  taken  not  to 
destroy  the  equation,  by  rendering  the  two  members  une- 
qual.    Many  changes  may  be  made  in  the  arrangement  of 
the  terms,  without  affecting  the  equality  of  the  sides. 

170.  Tke  reduction  of  an  equation  consists,  then,  in  bring- 
ing the  unknown  .quantity  by  itself,  on  one  side,   and  all  the 
known  quantities  on  the  other  side,  without  destroying  the  equa- 
tion. 

To  effect  this,  it  is  evident  that  one  of  the  members  must 
be  as  much  increased  or  diminished  as  the  other.    If  a  quan- 
tity be  added  to  one,  and  not  td  the  other,  the  equality  will 
be  destroyed.     But  the  members  will  remain  equal ; 
If  the  same  or  equal  quantities  be  added  to  each.  Ax.  1. 
If  the  same  or  equal  quantities  be  subtracted  from  each.  Ax.  2. 


SIMPLE  EQUATIONS.  73 

If  each  be  multiplied  by  the  same  or  equal  quantities.  Ax.  3. 
If  each  be  divided  by  the  same  or  equal  quantities.  Ax.  4. 

171.  It  may  be  farther  observed  that,  in  genera],  if  the 
unknown  quantity  is  connected  with  others  by  addition,  mul- 
tiplication, division,  &c.  the  reduction  is  made  by  a  contrary 
process.  If  a  known  quantity  is  added  to  the  unknown,  the 
equation  is  reduced  by  subtraction.  If  one  is  multiplied  by 
the  other,  the  reduction  is  effected  by  division,  &c.  The  rea- 
son of  this  will  be  seen,  by  attending  to  the  several  cases  in 
the  following  articles.  The  known  quantities  may  be  ex- 
pressed either  by  letters  or  figures.  The  unlcnoum  quantity 
is  represented  by  one  of  the  last  letters  of  the  alphabet,  gen- 
erally x,  y,  or  z.  (Art.  27.)  The  principal  reductions  to 
be  considered  in  this  section,  are  those  which  are  effected 
by  transposition,  multiplication,  and  division.  These  ought 
to  be  made  perfectly  familiar,  as  one  or  more  of  them 
will  be  necessary,  in  the  resolution  of  almost  every  equa- 
tion. 


TRANSPOSITION. 

172.  In  the  equation 

o?-7=9, 

the  number  7  being  connected  with  the  unknown  quantity  £ 
by  the  sign  —  ,  the  one  is  subtracted  from  the  other.  To 
reduce  the  equation  by  a  contrary  process,  let  7  be  added  to 
both  sides.  It  then  becomes 

x—  7  +  7=9  +  7. 

The  equality  of  the  members  is  preserved,  because  one  is  as 
much  increased  as  the  other.  (Axiom  1.)  But  on  one  side, 
we  have  —7  and  +7.  As  these  are  equal,  and  have  contra- 
ry signs,  they  balance  each  other,  and  may  be  cancelled.  (Art. 
77.)  The  equation  will  then  be 


Here  the  value  of  x  is  found.  It  is  shown  to  be  equal  to 
9  +  7,  that  is  to  16.  The  equation  is  therefore  reduced.  The 
unknown  quantity  is  on  one  side  by  itself,  and  all  the  known 
quantities  on  the  other  side. 

In  the  same  manner  if  x—b=a 

Adding  b  to  both  sides  x—  b-\-b=a+b 

And  cancelling  (  —  b  +  b)  x=a+b. 

Here  it  will  be  seen-  that  the  last  equation  is  the  same  as 


80  ALGEBRA. 

the  first,  except  that  6  is  on  the  opposite  side,  with  a  con- 
trary sign. 
Next  suppose 

Here  c  is  added  to  the  unknown  quantity  y.  To  reduce  the- 
equation  by  a  contrary  process,  let  c  be  subtracted  from  both 
sides,  that  is,  let  — c  be  applied  to  both  sides.  We  then  hare 

y+c— c—d— c 

The  equality  of  the  members  is  not  affected,  because  one  is 
as  much  diminished  as  the  other.  (Ax.  2.)  When  (+c— c) 
is  cancelled,  the  equation  is  reduced,  and  is  . 

y=d-c. 

This  is  the  same  as  y+c=rf,  except  that  c  has  been 
transposed,  and  has  received  a  contrary  sign.  We  hence 
obtain  the  following  general  rule  : 

173.  When  known  quantities  are  connected  uith  the  unknown 
quantity  by  the  sign  +  or  — ,  the  equation  is  reduced  by 
TRANSPOSING  the  known  quantities  to  the  other  side,  and  pre- 
fixing the  contrary  sign. 

This  is  called  reducing  an  equation  by  addition  or  subtrac- 
tion, because  it  is,  in  effect,  adding  or  subtracting  certain 
quantities,  to  or  from,  each  of  the  members. 

Ex.  1.  Reduce  the  equation  a: +36— m—h— d 

Transposing  +36,  we  hav.e  x— m=h  —  d— 3b 

And  transposing  —  w,  £=A— d— 36+ wi 

174.  When  several  terms  on  the  same  side  of  an  equation  are 
alike,  they  may  be  united  in  one,  by  the  rules  for  reduction  m 
addition.  (Art.  72  and  74.) 

Ex.  2.  Reduce  the  equation  ar+56— 47«=76 

Transposing  56— 4A  a?=76  —  56  +  4A 

Uniting  76—56  in  one  term         £=26+4A 

175.  The  unknown  quantity  must  also  be  transposed, 
whenever  it  is  on  both  sides  of  the  equation.  It  is  not 
material  on  which  side  it  is  finally  placed.  For  if  x =3; 
it  is  evident  that  3=x.  It  may  be  well  however,  t<r 
bring  it  on  that  side,  where  it  will  have  the  the  affirma- 
tive sign,  when  the  equation  is  reduced. 


SIMPLE  EQUATIONS.  81 

Ex.  3.  Reduce  the  equation  2x+2h=h  +  d+3x 

By  transposition  2h—h—d=z2x—2x 

Uniting  terms  h—d=x. 

176.  When  the  same  term  is  on  opposite  sides  of  the  equa- 
tion, instead  of  transposing,  we  may  expunge  it  from  each. 
For  this  is  only  subtracting  the  same  quantity  from  equal 
quantities.  (Ax.  2.) 


Ex.  4.  Reduce  the  equation 

Expunging  3A  x+d= 

Transp.  and  uniting  terms       x=b  +  6d. 

111.  As  all  the  terms  of  an  equation  may  be  transposed, 
or  supposed  to  be  transposed  ;  and  it  is  immaterial  which 
member  is  written  first;  it  is  evident  that  the  signs  of  all  the 
terms  may  be  changed,  without  affecting  the  equality. 

Thus,  if  we  have  x—b=d—a 

Then  by  transposition  —  d+a  =  —  x+b 

Or,  inverting  the  members  —  £+&  =  —  d+a. 

178.  If  all  the  terms  on  one  side  of  an  equation  be  trans- 
posed, each  member  will  be  equal  to  0. 

Thus  if  x+b=d,  .       then  x+b-d=0. 

It  is-  frequently  convenient  to  reduce  an  equation  to  this? 
form,  in  which  the  positive  and  negative  terms  balance  each 
other.  Ij  the  example  just  given,  x+b  is  balanced  by  —  «?. 
For  in  the  first  of  the  two  equations,  x-\-b  is  equal  to  J. 

Ex.  5.  Reduce  a+2*-8=J-4+*+a,  r\- 
Q.  Reduce  y+ab—  hm~a+2y—  ab-\-  hm. 
7.  Reduce  h+SO+lx^-Gh+Gx-d+b. 
&  Reduce  bh-21-Ax+d=12,-3x+d+lbh. 

£~  ?  -*• 

REDUCTION  or  EQUATIONS  BY  MULTIPLICATION. 

179.  The  unknown  quantity,  instead  of  being  connected 
with  a  known  quantity  by  the  sign  +  or  —,  may  be  divided 
by  it,  as  in  the  equation 

£    V 
^  * 


92  ALGEBRA. 

Here  the  reduction  can  not  be  made,  as  in  the  preceding 
instances,  by  transposition.  But  if  both  members  be  multi- 
plied by  a,  (Art.  170.)  the  equation  will  become 

x  =ab. 

For  a  fraction  is  multiplied  into  its  denominator,  by  remov- 
ing the  denominator.  This  has  been  proved  from  the  prop- 
erties of  fractions.  (Art.  159.)  It  is  also  evident  from  the 
sixth  axiom. 

ax     3a?     (a+b)xx     dx+5x 

Thus  a?=  —  =~^—  -  TA  —  =~TTT~>  &£•     F°r  m  eac^ 
a      o          a-\-o  a+o  ' 

of  these  instances,  x  is  both  multiplied  and  divided  by  the 
same  quantity;  and  this  makes  no  alteration  in  the  value. 
Hence, 

180.  When  the  unknoivn  quantity  is  DIVIDED  by  a  known 
quantity,  the  equation  is  reduced  by  MULTIPLYING  each  side  by 
this  known  quantity. 

The  same  transpositions  are  to  be  made  in  this  case,  as  in 
the  preceding  examples.  It  must  be  observed  also,  that 
every  term  of  the  equation  is  to  be  multiplied.  For  the  sev- 
eral terms  in  each  member  constitute  a  compound  multipli- 
cand, which  is  to  be  multiplied  according  to  art.  98. 

<y* 

Ex.  1.  Reduce  the  equation  —  +a=xb-\-d 


Multiplying  both  sides  by 


The  product  is  x+ac=bc+cd 

And  transposing  ac  x=bc+cd—ac. 

a?— 4  :« 

2.  Reduce  the  equation  -      -  -{-5=20 

Multiplying  by  6  #-4+30  =  120 

Transp.  and  uniting  terms-        * =120 +4  —  30=94. 

x 
2.  Reduce  the  equation  ——-r+d^h 

Multiplying  by  «+  6  (Art.  100.)    x+ad+bd=ah  +  bh 
By  transposition  x=ah-{-bh—ad—bd. 

,181.  When  the  unknown  quantity  is  in  the  denominator  of 
a  fraction,  the  reduction  is  made  in  a  similar  manner,  by  mul- 
tiplying the  equation  by  this  denominator. 


SIMPLE  EQUATIONS.  S3 

6 
Ex.  4.  Reduce  the  equation  To^c"*"'7==^ 

Multiplying  by  1 0 — x  6  +  70 — lx — 80 — 8x 

Transp.  and  uniting  terras          a: =4. 

182.  Though  it  is  not  generally  necessary,  yet  it  is  often 
convenient,  to  remove  the  denominator  from  a  fraction  con- 
sisting of  known  quantities  only.     This  may  be  done,  in  the 
same  manner,  as  the  denominator  is  removed  from  a  fraction 
which  contains  the  unknown  quantity. 

x      d     ft 
Take  for -example  — =-T-+  — 

ad     ah 
Multiplying  by  a  (Art.  1 58.)  x = y  -f — 

abli 
Multiplying  by  b  +x=ad+ 

Multiplying  by  c  bcx=-acd+abh. 

Or  we  may  multiply  by  the  product  of  all  the  denomina- 
tors at  once. 

x      d      k 
In  the  same  equation  — =~Z~+ — 

abcx     abed     dbch 
Multiplying  by  abc  ~a~=~b~'ir — ~ 

Then  by  cancelling  from  each  term,  the  letter  which  is 
common  to  its  numerator  and  denominator,  (Art.  145,)  we 
have  bcx—acd+abh,  as  be- 

fore.    Hence, 

183.  Jin  equation  may  be  cleared  of  FRACTIONS  by  multiply- 
ing each  side  into  all  the  DENOMINATORS. 

x      b       e       h 

Thus  the  equation  — —~j+ — — — 

a      a      g     m 

Is  the  same  as  dgmx  —abgm  +  adem—aJgh. 

*•      2      4      6 

And  the  equation  ~2~~3~^~~5~^~2 

Is  the  same  as  30* =40 +48  + 180. 

* 

REDUCTION  or  EQUATIONS  BY  DIVISION. 

184.  When  the  unknown  quantity  is  MULTIPLIED  into  any 
Jcnown  quantity,  the  equation  is  reduced  by  DIVIDING  both  sid&r 
•Jty  this  known  quantity.  (Ax.  4.) 


ALGEBRA. 


Ex.  1.  Reduce  the  equation  ax+b—  3A=rf 

By  transposition  ax=d+3k—  b 

n    -A-      u  d+3h-b 

Dividing  by  a  #  =  --  ' 

a      d 

2.  Reduce  the  equation  2xi=—  —  -r 

c       /t 

Clearing  of  fractions  (Art.  183.)  2chx=ah  — 

ah—cd+tibch 
Dividing  by  2cA  x—  --  ^T  -- 

185.  If  the  unknown  quantity  has  co-efficients  in  several 
terms,  the  equation  must  be  divided  by  all  these  co-efficients, 
connected  by  their  signs,  according  to  art.  121. 

' 

Ex.  3.  Reduce  the  equation  3x-\-d=bx+a 

By  transposition  3x—bx=a—d 

That  is,  (Art.  120.)  (3-6)  X  x—a^d 

n—d 
Dividing  by  3  —  b  X=^~T 

Ex.  4.  Reduce  the  equation  cw-f4=A—  .v 

By  transposition  ax-\-  x=h—  4 

That  is  (a-t-l)xx=h—4 

A-4 
Dividing  by  a  +  1  x  = 


x—  b 

Ex.  5.  Reduce  the  equatien  x~~~fi  — 

Clearing  of  fractions.  See  art.  142.  Ahx  —  (Ax—  Ab)=a 
That  is  (Art.  82.)  4Ax—  4^+46  =ah+dh 

Transposing  46  4/w—  4.xi=0A+c?A—  46 

ah  +  dh—  46 
Dividing  by  4A—  4  x=  —  .  ,  _^  — 

186.  If  any  quantity,  either  known  or  unknown,  is  found 
as  a  factor  in  every  term,  the  equation  may  be  divided  by  it. 
On  the  other  hand,  if  any  quantity  is  a  divisor  in  every  term, 
the  equation  may  be  multiplied  by  it.  In  this  way,  the  fac- 
jLpr  or  divisor  will  be  removed,  so  as  to  render  the  expression 
more  simple. 


SIMPLE  EQUATIONS.  65 

Ex.  6.  Reduce  the  equation  ax+3ab=Qad-\-a 

Dividing  by  a  (Arts.  120  and  128.)    x+3b=6d+l 
And  transposing  36  x=6d+I  —  36 


A'-fl      $      h— d 

X  X~     X 


7.  Reduce  the  equation 

Multiplying  by  x  (Art.  159.)  x+  1  —b—h—d 

And  transposing  1  —6  x=h—  d+b  —  1. 


8.  Reduce  the  equation  #X  (a+6)—  a+6=rfx  (a+b) 

Dividing  by  a  +  b  (Art.  118.)*  —  l—d 
Transposing  —1  x 


187.  Sometimes  the  conditions  of  a  problem  are  at  first 
stated,  not  in  an  equation,  but  by  means  of  ^proportion.   To 
show  how  this  may  be  reduced  to  an  equation,  it  will  be  ne- 
cessary to  anticipate  the  subject  of  a  future  section,  so  far  as 
to  admit  the  principle,  that  "  when  four  quantities  are  in  ge- 
ometrical proportion,  the  product  of  the  two  extremes  is 
equal  to  the  product  of  the  two  means :"  a  principle  which 
is  at  the  foundation  of  the  Rule  of  Three  in  arithmetic.   See 
Webber's  Arithmetic.  < 

Thus  if  a :  b : :  c :  d,  Then  ad= be 

And  if  3:4::6:8;  And   3x8=4x6.    Hence, 

188.  A  proportion  is  converted  into  an  equation,  by  making 
the  product  of  the  extremes,  one  side  of  the  equation ;  and  the 
product  of  the  means,  the  other  side. 

Ex.  1 .  Reduce  to  an  equation  ax  :b::ch:d. 

The  product  of  the  extremes  is  adx 

The  product  of  the  means  is  bch 

The  equation  is,  therefore  adx=bch. 

2.  Reduce  to  an  equation  a+b:c::h  — miy. 

The  product  of  the  extremes  is  ay+by 

The  product  of  the  means  is  ch—cm 

The  equation  is,  therefore  ay-\-by=ch—cm. 

189.  On  the  other  hand,  an  equation  may  be  converted  into 
a.  proportion,  by  resolving  one  side  of  the  equation  into  two 

factors,  for  the  middle  terms  of  the  proportion;  and  the  other 
into  two  factors,  for  the,  extremes. 


Sft  ALGEBRA. 

As  a  quantity  may  often  be  resolved  into  different  pairs  of 
/actors;  (Art.  42.)  a  variety  of  proportions  may  frequently 
be  derived  from  the  same  equation. 

Ex.  1.  Reduce  to  a  proportion  abc=deh 

The  side  nbc  may  br  resolved  into  a  x  be,  or  ab  x  c,  or  ac  X  b. 
And  de.h  may  be  resolved  into          dx  eh,  or  de  x  h  or  dh  X  e. 
Therefore  a:d::eh:bc  And  ac:dh::e:b 

And  ab:dc::h:c  And  ac : f? : : e^t : b &c. 

For  in  each  of  these  instances,  the  product  of  the  ex- 
tremes is  oic,  and  the  product  of  the  means  deh. 

2.  Reduce  to  a  proportion  ax+bx=cd— ch 

The  first  member  may  be  resolved  into          x  x  (a+b) 
And  the  second  into  c  x  (d—h) 

Therefore  x:c:: d—h  :  a -j- b         And  d — h  :x::a  +  b:c,  &c. 

190.  If  for  any  term  or  terms  in  an  equation,  any  other 
•expression  of  the  same  value  be  substituted,  it  is  manifest 
that  the  equality  of  4he  sides  will  not  be  affected. 

64  . 
Thus,  instead  of  16,  we  may  write  2x  8  or  -j-,  or  25— 9,  See. 

For  these  are  only  different  forms  of  expression  for  the 
same  quantity. 

191.  It  will  generally  be  well  to  have  the  several  steps,  in 
the  reduction  of  equations,  succeed  each  other  in  the  follow- 
ing order. 

First,  Clear  the  equation  of  fractions.  (Art.  183.) 
Secondly,  Transpose  and  unite  the  terms.  (Arts.173-,  4,  5.) 
Thirdly,  Divide  by  the  co-efficients  of  the  unknown  quan- 
tity. (Arts.  184,  5.) 


EXAMPLES. 

$x          Sx 
,  Reduce  the  equation  — -j-6=-^-+7 

Clearing  of  fractions  24v  + 1 92 = 20* + 221 

Transp.  and  uniting  terms  4.* =32 

Dividing  by  4  x =8. 


SIMPLE  EQUATIONS.  87 

X  X         X 

%.  Reduce  the  equation  — 4-A=-r-— — -fj 

ft  0         C 

Clearing  of  fractions  bcx+abch=acx— abx-\-  abed 

By  transposition  bcx+abx  —  acx=abcd—abck 

abed — abch 

Dividing  by  the  co-efFs  of  x.   x=7 — ; — i 

&   J  bc-\-ab—ac 

3.  Reduce  40— 6*—16=rl20— 14av  Ans.  #  =  12. 

*-3     x  *-19  93 

4.  Reduce  — —  +-5=20— — 5—  Ans.  x=-r-. 

&          o  &  Qc          t 

xx  x  I— a 

5.  Reduce  -5  +  -F=20— -.'          6.  Reduce -4=5. 

Q        O  4-        ;  X 

3  6x 

7.  Reduce  -_^-2=8.  /.    8.   Reduce  ^-L 


SOLUTION   OF   PROBLEMS, 

192.  In  the  solution  of  problems,  by  means  of  equations, 
tvvo  things  are  necessary  :  First  to  translate  the  statement 
of  the  question  from  common  to  algebraic  language,  in  such 
a  manner  as  to  form  an  equation :  Secondly,  to  reduce  this 
equation  to  a  state  in  which  the  unknown  quantity  will  stand 
by  itself,  and  its  value  be  given  in  known  terms,  on  the  oppo- 
site side.     The  manner  in  which  the  latter  is  effected,  has  al- 
ready been  considered.     The  former  will  probably  occasion 
more  perplexity  to  a  beginner ;  because  the  conditions  of 
questions  are  so  various  in  their  nature,  that  the  proper  me- 
thod of  stating  them  can  not  be  easily  learned,  like  the  re- 
duction of  equations>  by  a  system  of  definite  rules.      Prac- 
tice however  will  soon  remove  a  great  part  of  the  difficulty. 

193.  It  is  one  of  the  principal  peculiarities  of  an  algebra- 
ic solution,  that  the  quantity  sought  is  itself  introduced  into- 
the  operation.     This  enables  us  to  make  a  statement  of  the 
conditions,  in  the  same  form,  as  though  th^  problem  were  al- 
ready solved.     Nothing  then  remains  to  be  done,  but  to  re- 
duce  the   equation,  and  to  find  the  aggregate  value  of  the. 
known  quantities.  (Art.  53.)      As  these  are  equal  to  the  un- 
known quantity  on  the  other  side  of  the  equation,  the  value 


83  ALGEBRA. 

of  that  also  is  determined,  and  therefore  the  problem  is 
solved. 

Problem  1.  A  man  being  asked  how  much  he  gave  for  his 
watch,  replied  ;  If  you  multiply  the  price  by  4,  and  to  the 
product  add  70,  and  from  this  sum  subtract  50,  the  remain- 
der will  be  equal  to  220  dollars. 

To  solve  this,  we  must  first  translate  the  conditions  of  the 
problem,  into  such  algebraic  expressions  ,as  will  form  an 
equation. 

Let  the  price  of  the  watch  be  represented  by        x 
This  price  is  to  be  mult'd  by  4,  which  makes        Ax 
To  the  product,  70  is  to  be  added,  making         4^+70 
From  this,  50  is  to  be  subtracted,  making  4#+70 — 50 

Here  we  have  a  number  of  the  conditions,  expressed  N  in 
algebraic  terms;  but  have  as  yet  no  equation.  We  must  ob- 
serve then,  that  by  the  last  condition  of  the  problem,  the 
preceding  terms  are  said  to  be  equal  to  220. 

We  have,  therefore,  this  equation  4* +70— 50=220. 
To  reduce  this, 

Transpose  and  unite  the  terms,  then       4.x1  =200 

Divide  by  4  (Art.  184.)  and  #=50. 

Here  the  value  of  x  is  found  to  be  50  dollars,  which  is  tho 
price  of  the  watch. 

194.  To  prove  whether  we  have  obtained  the  true  value 
of  the  letter  which  represents  the  unknown  quantity,  we  have 
only  to  substitute  this  value,  for  the  letter  itself,  in  the  equa- 
tion which  contains  the  first  statement  of  the  conditions  of 
the  problem ;  and  to  see  whether  the  sides  are  equal,  after 
the  substitution  is  made.  For  if  the  answer  thus  satisfies 
the  conditions  proposed,  it  is  the  quantity  sought.  Thus,  in 
the  preceding  example, 

The  original  equation  is  4.v+ 70— 50=220 

Substituting  50  for  a;,  it  becomes      4  x  50  +  70—50=220. 

To  see  whether  the  first  member  of  this  equation  is  equal 
to  the  second, 

Multiply  4  into  50 ;  the  product  is  200 

To  this  add  70 


The  sum  is  .270 

From  this  subtract  50 

The  remainder  is  220  as  in 
the  statement  of  the  problem. 


SIMPLE  EQUATIONS.  89 

Prob.  2.  What  number  is  that,  to  which  if  its  half  be 
added,  and  from  the  sum  20  be  subtracted,  th©  remainder 
will  be  a  fourth  part  of  the  number  itself? 

In  stating  questions  of  this  kind,  where  fractions  are  con- 
cerned, it  should  be  recollected,  that  ±x  is  the  same  as 
x  2.x 

—  ;  that  !#=y,  &e-  (Art.  161.) 

In  this  problem,  let  x  be  put  for  the  number  required. 

x  x 


Then  by  the  conditions  proposed,  x+-^  —  20=-j 

Clearing  of  fractions  8x+Ax—.  160  =2x 

Transp.  and  uniting  terms  10.*  =  160 

Dividing  by  10  #=16. 

16  18 

Proof 


, 

Prob.  3.  A  father  divides  his  estate  among  his  three  sons, 
in  such  a  manner,  that, 

The  first  has  $1000  less  than  half  of  the  whole  j 
The  second  has  800  less  than  a  third  of  the  whole  j 
The  third  has  600  less  than  a  fourth  of  the  whole  j 
What  is  the  value  of  the  estate  ? 
If  the  whole  estate  be  represented  by  x,  then  the  several 

XXX 

shares  will  be  -^— lOO^and  17  — SOO^and  -r—  600? 

And  as  these  constitute  the  whole  estate,  they  are  togeth- 
er equal  to  x. 

X  XX 

We  have  then  this  equation  IT  —  1 000+^-800  +  -r-QOO=x 

**  J  4 

Or,  by  uniting  terms,  -«"  -t-"or-f--r  — 2400=*- 

Clearing  of  fractions  I2x + 8x + 6*  -  57600 = 24* 

Transp.  and  uniting  terms       2^=57600 
Dividing  by  2  x— 28800 

28800  28800  28800 

Proof,  — ^—  - 1 000 + — g  —  —  800  +  —£—  ~  600 = 28SOO, 

195.  To  avoid  an  unnecessary  introduction  of  unknown 
quantities  into  an  equation,  it  may 'be  well  to  observe,  in  this 
place,  that  when  the  sum  or  difference  of  two  quantities  is 
M 


Uf  ALGEBRA. 

given,  both  of  them  may  be  expressed  by  means  of  tu« 
same  letter.  For  if  one  of  two  quantities  be  subtracted 
from  their  sum,  it  is  evident  the  remainder  will  be  equal  to 
the  other.  And  if  the  difference  of  two  quantities  be  sub- 
tracted from  the  greater,  the  remainder  will  be  the  less. 
Thus  if  the  sum  of  two  numbers  be  20 

And  if  one  of  them  be  represented  by  x 

The  other  will  be  equal  to  20—*, 

'• 

Prob.  4.  Divide  48  into  two  such  parts,  that  if  the  less  be 
divided  by  4  and  the  greater  by  6,  the  sum  of  the  quotients 
will  be  9. 

Here  if  A;  be  put  for  the  smaller  part,  the  greater  will  be 
48-*. 

x     48—  x 
By  the  conditions  of  the  problem  ~r+  —  g  —  =9. 

Clearing  of  fractions  6*+  192—  4*=21S 

Transposing  and  uniting  terms  2#=24 

Dividing  by  2  ^=12,  the  less. 

Then  48  -*=48-  12=36,  the  greater. 

196.  Letters  may  be  employed  to  express  the  known 
quantities  in  an  equation,  as  well  as  the  unknown.  A  par- 
ticular value  is  assigned  to  the  numbers,  when  they  are  intVo- 
duced  into  the  calculation:  and  at  the  close,  the  numbers 
are  restored.  (Art.  52.) 

Prob.  5.  If,  to  a  certasr  number,  720  be  added,  and  the 
sum  be  divided  by  125  ;  the  quotient  will  be  equal  to  7392 
divided  by  462.  What  is  that  number  ? 

Let  x=  the  number  required* 
a  =720  rf 

£=125  A  3=48*2' 

x+a     d 
Then  by  the  conditions  of  the  problem  —  —  =  - 


Clearing  of  fractions 

Transposing  ah  hx=bd—ak 

,  bd—fth 

Dividing  by  h  x  =  —  T~ 

(125x7392)-(720x462) 
Restoring  the  numbers,  x  =  ~~ 


SIMPLE  EQUATIONS.  91 

197.  When  the  resolution  of  an  equation  brings  out  a  neg- 
ative answer,  it  shows  that  the  value  of  the  unknown  quanti- 
ty is  contrary  to  the  quantities  which,  in  the  statement  o.f  the 
questjon,  are  considered  positive.  See  Negative  Quantities. 
(Art.  54,  &c.) 

Prob.  6.  A  merchant  .gains  or  loses,  in  a  bargain,  a  certain 
-sum.  In  a  second  bargain,  he  gains  350  dollars,  and,  in  a 
third,  loses  GO.  In  the  end,  he  finds  he  has  gained  200  dol- 
lars, by  the  three  together.  How  much  did  he  gain  or  lose 
by  the  first? 

In  this  example,  as  the  profit  and  loss  are  opposite  in  their 
nature,  they  must  be  distinguished  by  contrary  signs.  (Art. 
£7.)  If  the  profit  is  marked  -f ,  the  loss  must  be  — . 

Let  x—  the  sum  required. 

Then  according  to  the  statement  #+350  —  60=200 

By  transposition  #=200+60— 350 

And  uniting  the  terms  x—  —  90. 

\ 

The  negative  sign  prefixed  to  the  answer,  shows  that  there 
was  a  loss  in  the  first  bargain ;  and  therefore  that  th«  proper 
sign  of  x  is  negative  also.  But  this  being  determined  by  tho 
answer,  the  omission  of  it  in  the  course  of  the  calculation 
can  lead  to  no  mistajce. 

Prob.  7.  A  ship  sails  4  degrees  north,  then  13  S.  then  17 
N.  tb.cn  19  S.  and  has  finally  11  degrees  of  south  latitude. 

What  was  her  latitude  at  starting  ? 

/ 

Let  x  =  the  latitude  sought. 
Then  marking  the  northings  + ,  and  the  southings  —  ; 

By  the  statement  *+4— 13+17  — 19  =  — 11 

By  transposition  ^  =  13+19  —  11—4  —  17 

And  uniting  terms  #=0. 

The  answer  here  shows  that  the  place  from  which  the  ship 
started  was  on  the  equator,  where  the  latitude  is  nothing. 

Prob.  8.  If  a  certain  number  is  divided  by  12,  the  quo- 
tient, dividend,  and  divisor  added  together  will  amount  .t« 
64.  What  is  the  number? 

Let  .v=r  the  number  sought. 


ALGEBRA. 


x 
Then  +x+  12=64 


Multiplying  by  12,  (Art.  180.)         x  +12*+!  44  =763 
Transposing  and  uniting  terms         13Jr=624 

624 
Dividing  by  13  x=-y^-=48. 

Prob.  9.  An   estate  is  divided  among  four  children, 
such  a  manner,  that 

The  first  has  200  dollars  more  than  £  of  the  whole, 
The  second  has  340  dollars  n;ore  than  |  of  the  whole, 
The  third  has  300  dollars  more  than  \  of  the«whole, 
The  fourth  has  400  dollars  more  than  \  of  the  whole. 
"What  is  the  value  of  the  estate?  Ans.  4300  dollars. 

.,     Prob.  10.  What  is  that  number  which  is  as  much 
ihan  500,  as  a  fifth  part  of  it  is  greater  than  40?  Ans.  450. 

Prob.  11.  There  are  two  numbers  whose  difference  is  40, 
and  which  are  to  each  other  as  6  to  5.  What  are  the  num- 
ber? ? 

jLet  x=z  the  greater. 

Then  x—  40=  the  less.  (Art.  195.) 

3y  the  conditions  of  the  question  x  :  x—  40  :  :6  :  5 

Mult'g  extremes  and  means  (Art.  188.)  6#—  240=5.v 
Transposing  and  uniting  terms  x=240,  the  greater. 

And  ^  ^^  240  -40  =200,  the  less. 

Prob.  12.  Three  persons,  A,  B,  and  C  draw*  prizes  in  a 
Jottery.  A  draws  200  dollars;  B  draws  as  much  as  A,  to- 
gether with  a  third  of  what  C"  draws  ;  and  C  draws  as  much 
AS  A  and  B  both.  What  is  the  amount  of  the  three  prizes. 

'  Ans-  120°  doUars- 


Prob.  13.  What  number  is  that,  which  is  to  12  increased 
by  three  times  the  number,  as  2  to  9  ?  Ans.  8. 

Prob.  14.  A  ship  and  a  boat  are  descending  a  river  at  the 
same  time.  The  ship  passes  a  certain  fort,  when  the  boat  is 
13  miles  below.  The  ship  descends  five  miles,  while  the 
boat  descends  three.  At  what  distance  below  th^  fort,  wiR 
:hey  be  together  ? 

Let  x=  the  distance  required. 


SIMPLE  EQUATIONS.  93 

Then  by  the  question  x :  x  —  1 3 : :  5 : 3 

Mult,  extremes  and  means  Sx— 65=3* 

Transp.  and  uniting  terms  2r=65 

Dividing  by  2  ^   *=32f 

Prob.  15.  Wbut  number  is  mat,  a  sixth  part  of  which  ex- 
ceeds an  eighth  part  of  it  byj20?f  ,  L  •  y  Ans.  450. 

•/-  Prob.  16.  Divme'  a  prize  of  2000  dollars  into  two  such 
parts,  that  one  of  them  shall  be  to  the  other,  as  9 : 7. 

..Ansj,  The  parts  are  1125,  and  875. 
X   ±-  jf  -/-.i  —  y<r 

Prob.  17.  Whm  sum  ormoney  is  that,  whose  third  part, 
fourth  part,  and  fifth  part,  added  together,  amount  to  94 
dollars?  Ans.  12,0  dollars. 

^  Prob.  18.  Two  travellers,  A  and  B,  360  miles  apart,  trav- 
el towards  each  other  till-  they  meet.  A's  progress  is  10 
miles  an  hour,  and  B's  8.  How  far  does  each  travel  before 
they  meet?  Ans.  A  goes  200  miles,  and  B  160. 

Prob.  19.  A  man  spent  one  third  of  his  life  in  England, 
one  fourth  of  it  in  Scotland,  and  the  remainder  of  it,  which 
was  20  years,  in  tfce  United  States.  To  what  age  did  he 
live  ?  Ans.  To  the  age  of  48. 


For  the  solution  of  many  algebraic  problems,  an  acquaint- 
ance with  the  calculations  of  powers  and  radical  quantities  is 
required.  It  will  therefore  be  necessary  to  attend  to  these-, 
befq? e  finishing  the  subject  of  equations. 


SECTION  VIII. 


INVOLUTION  AND  POWERS. 


\RT    198  a  quantity  is  multiplied  into  ITSELF, 

the  PRODUCT  is  called  a  POWER. 

Thus  2x2=4,    the  square  or  second  power  of  2. 

2x2x2=8,    the  cube  or  third  power. 
2  x  2  x  2  x  2 = 1 6,  the  fourth  power,  Sic. 

So  10  x  10  =  100,      the  second  power  of  10. 

10x10x10  =  1 000,    the  third  power. 
10  x  10  x  10  x  10=10000,  the  fourth  power,  &c. 

And  cxfl=ffl«,      the  second  power  of  «. 

«  X  a  x  a=aaa,    the  third  power. 
axaxa xa=aaaa,  the  fourth  power,  &c. 

199.  The  original  quantity  itself,  though  not,  like   the 
powers  proceeding  from  it,  produced  by  multiplication,  is 
nevertheless  called  the  first  power.      It  is  also  called  the  root 
•f  the  other  powers,  because  it  is  that  from  which  they  are 
all  derived. 

200.  As  it  is  inconvenient,  especially  in  the  case  of  high 
powers  to  write  down  all  the  letters  or  factors  of  which  the 
powers  are  composed,  an  abridged  method  of  notation  is 
generally  adopted.     The  root  is  written  only  once ;  and  then 
a  number  or  letter  is  placed  at  the  right  hand,  and  a  little  el- 
evated, to  signify  how  many  times  the  root  is  employed  as  a 

factor,  to  produce  the  power.  This  number  or  letter  is  call- 
ed the  index  or  exponent  of  the  power.  Thus  a2  is  put  for 
a  X  a.  or  aa,  because  the  root  a  is  turice  repeated  as  a  factor, 
to  produce  the  power  aa.  And  a3  stands  for  aaa ;  for  here 
a  is  repeated  three  times  as  a  factor. 

The  index^of  ihe  first  power  is  1 ;  but  this  is  commonly 
omitted.     Thus  a1  is  the  same  as  a. 

201.  Exponents  must  not  be  confounded  with  co-efficients. 


POWERS.  95 

A  co-efficient  shows  how  often  a  quantity  is  taken  as  a  part 
of  a  whole.  An  exponent  shows  how  often  a  quantity  is  ta- 
ken as  a.  factor  in  a  product. 

Thus4a=a+a+a4-a.  But  a4=axaxaxa. 

202.  The  scheme  of  notation  by  exponents  has  the  pe- 
culiar advantage  of  enabling  us  to  express  art  unknown  pow- 
er.    For  this  purpose  the  index  te  a  letter,  instead  of  a  nu- 
meral figure.       In  the  solution  of  a  problem,  a  quantity 
may  occur,  which  we  know  to  be  some  power  of  another 
quantity.     But  it  may  not  be  yet  ascertained  whether  it  i» 
a  square,  a  cube,  or  some  higher  power.      Thus  in  the  ex- 
pressioa  cr%  the  index  x  denotes  that  a  is  involved  to  some 
power,  though  it  does  not  determine  what  power.     So  bm, 
and  dn  are  powers  of  b  and  d;  and  are  read  the  with  power 
of  b,  and  the  nth  power  of  d.     When  the  value  of  the  index 
is  found,  a  number  is  generally  substituted  for  the  letter. 
Th«s  if  ro=3,  then  bm=b3  :  but  if  m==5,  then  bm=bs. 

203.  The  method  of  expressing  powers  by  exponents  is 
also  of  great  advantage  in  the  case  of  compound  quantities. 
ThUs 


X(a+b+d)  that  is,  the  cube  of  (a+b+d).      But  this  in- 
volved at  length  would  be  [d* 


204.  If  we  take  a  series*  of  powers  whose  indices  increase 
er  decrease  by  1,  we  shall  find  that  the  powers  themselves 
increase  by  a  common  multiplier,  or  decrease  by  a  common 
divisor  ;  and  that  this  multiplier  or  divisor  is  the  original 
quantity  from  which  the  powers  are  raised. 

Thus  in  the  series  aaaua,       aaaa,       aaa,       aa,       a  ; 

Or  a*  a*  a3         a*      a1  ; 

the  indices  counted  from  right  to  left  are  1,  2,  3,  4,  5  ;  and 
the  common  difference  between  them  is  a  unit.  If  we  begin 
on  the  right,  and  multiply  by  a,  we  produce  the  several  pow- 
ers, in  succession,  from  right  to  left. 

Thus  «xa=a2  the  2d  term.          And  a3  xa  =  fl*. 
a2  Xa=a3  the  3d  term.  a4  xa=a5,  Jo\ 

If  we  begin  on  the  left,  and  divide  by  a, 

*NoTK.  The  term  scries  is  applied  to  a  number  of  quantities  sue 
ceeding  each  other,  in  some  regular  order.  It  is  nqtto£0ja£ned  to  a;jy 
particular  law  of  increase  or  decrease. 


96  ALGEBRA. 

We  have  o'4-a=a*.  And  «3-^«=a>. 

a«-~a=a3.  a2-=-a=al. 

205.  But  this  division  may  be  carried  still  farther  j  ami 
we  shall  then  obtain  a  new  set  of  quantities. 

Thus  a~«=—  =1.  (Art.  128.)  And  —  ~«=  —  (Art.163.) 
a  a  aa  ^ 


The  whole  series  then 

1      1         1 

Is  aaaaa.  aaaa.  aaa.  aa,  a.  1,  —  >    —  >    -  >    &c. 

'  a     aa,      maa 

Or   a',     a4,    a3,  a2,  a,  1,—  >    ^>    ^y»      &c. 

Here  the  quantities  on  the  right  of  1,  are  the  reciprocals 
of  those  on  the  left.  (Art.  49.)  The  former,  therefore,  may 
be  properly  called  reciprocal  powers  of  a;  while  the  latter 
may  be  termed,  for  distinction  sake,  direct  powers  of  a.  It 
may  be  added,  that  the  powers  on  the  left  are  also  the  recip- 
rocals of  those  on  the  right. 

For  1-r—  =lxy=a.  (Art.  162.)     And  l+-j=a*. 
1  a1  1 


206.  The  same  plan  of  notation  is  applicable  to  compound 
quantities.     Thus,  from  a-f&,  we  have  the  series, 


207.  For  the  convenience  of  calculation,  another  form  of 
notation  is  given  to  reciprocal  powers. 

11  11 

According  to  this,  —  or-7=  a     '    And  --  or~j=a~". 
'a      a1  aaa      a3 

1  1  _  11 


.  4 

aa     a*  aaaa     a 


And  to  make  the  indices  a  complete  series,  with  I  for  the 


INVOLUTION.  97 

common  difference,  the  term —  or  1,  which  is  censidered  as 

no  power,  is  written  a°. 

The  powers  both  direct  and  reciprocal*  then, 

a      1       1        1  1 

Instead  of  aaaa,  ««a,  a*,  a,  ~>    ~»    ->    ~->    —>  &c. 

Will  be  a4,    a3    aa,a1,a°,a-1,a-2,a-3,      a~4,  &c. 

Or  a+4,a+3,  a+3,a+1,ao, a"1, «-»,«-»,      a~4,  fcc. 

And  the  indices  taken  by  themselves  will  be, 

+4,  +3,  +2,  +1,  0,  -1,  -2,  -3,  -4,  Sic. 

• 

208.  The  root  of  a  power  may  be  expressed  by  more  let- 
ters than  one. 

Thus  aaxaa,  or  aa\     is  the  second  power  of  aa. 

And  aa  x  aa  x  aa,  or  aa|      is  the  third  power  of  aa,  &c. 

Hence  a  certain  power  of  one  quantity,  may  be  a  differ- 
ent power  of  another  quantity.  Thus  a4  is  the  second  pow- 
er of  a2,  and  the  fourth  power  of  a. 

209.  All  the  powers  of  1  are  the  same.      For  1x1,  or 
lXlXl,&c.  is  still  1. 

INVOLUTION. 

210.  Involution  is  finding  any  power  of  &  quantity,  by 
multiplying  it  into  itself.     The  reason  of  the  following  gen- 
eral rule  is  manifest,  from  the  nature  of  powers. 

Multiply  the  quantity  into  itself,  till  it  is  taken  as  a  factor^ 
as  many  times  as  there  arc  units  in  the  index  of  the  power  to 
which  the  quantity  is  to  be  raised. 

This  rule  comprehends  all  the  instances  which  can  occur 
in  involution.  But  it  will  be  proper  to  give  an  explanation 
of  the  manner  in  which  it  is  applied  to  particular  cases. 

211.  A  single  letter  is  involved,  by  giving  it  the  index  of 
the  proposed  power;  or  by  repeating  it  as  many  times,  as 
there  are  units  in  that  index. 

The  4th  power  of  a,  is  a4  or  aaoa.  (Art.  198.) 

The  6th  power  of  y,  is  y6  or  yyyyyy* 

The  nth  power  of  x,  is  an  or  xxx .,.  n  times  repeated, 

*  See  note  B, 

N 


98  ALGEBRA. 

212.  The  method  of  involving  a  quantity  which  consists  of 
several  factors,  depends  on  the  principle,  that  the  power  cf 
the  product  of  several  factors  is  equal  to  the  product  of  their 


Thus  (cty)*=*a*y*.    For  hy  art.  210;  (ay}*  =ay  x  ay. 
But  ayxay=ayay=aayy—a2y*. 
So  (bmx)  3  =bmx  x  bnix  x  bmx—bbbmmmxxx=b3m9x9  . 
And  (ady}n  —ady  x  ady  x  ady  ...  n,  times  =an  d"  yn  . 

Tn  finding  the  power  of  a  product,  therefore,  we  may  ei- 
ther involve  the  whole  at  once  ;  or  we  may  involve  each  of 
the  factors  separately,  and  then  multiply  their  several  powers 
into  each  other. 

Ex.  1.  The  4th  power  of  dhy,  is  (dhy}*,  or  d*h4y4. 

2.  The  3d  .power  of  46,  is  (4£)3,  or  4363,  or  646  3. 

3.  The  nth  power  of  Gad,  is  (&ad}n  ,  or  6n  a"  d"  . 

4.  The  3d  power  of  3m  x  2y,  is  (3m  x  2y)  •  ,  or  27m  3  X  Qy  '  . 

213.  A  dbmpound  quantity,  consisting  of  terms  connect- 
ed by  +  and  —  ,  is  involved  by  an  actual  multiplication  of 
its  several  parts.  Thus, 

=a+6,  the  first  power. 

«2+  ab 
+  ab+l2 


(a+b)2=a*  +2a6+6z,  the  second  power  of  (a 
a  +b 


a  +  b 


ab* 
+&8 

2-{-&3,  the  3d  power. 


4,  the  4th power, 


INVOLUTION. 

2.  The  square  of  a—b,  is  a2—  2ab  +  bz. 

3.  The  cube  of  a+1,  is  o3+3ca+c>6 

4.  Thesquarea+&  +  A,isa 

5.  Required  the  cube  of  a 

6.  Required  the  4th  power  of  b  +  2. 

214.  The  squares  of  binomial  and  residual  quantities  oc- 
cur so  frequently  in  algebraic*  processes,  that  it  is  important 
to  make  them  familiar. 

If  we  multiply  «+A  into  itself,  and  also  a—  A, 

We  have  a+h  And  a  —  k 

«  —  h 


a*  -{-ah  a2  —ah 

+  ah+h*  -ah+h* 


a2+2«A+A2. 

Here  it  will  be  seen  that,  in  each  case,  the  first  and  last 
terms  are  squares  of  a  and  A ;  and  that  the  middle  term  is 
twice  the  product  of  a  into  A.  Hence  the  squares  of  bino- 
mial and  residual  quantities,  without  multiplying  each  of  the 
terms  separately,  may  be  found,  by  the  following  proposi- 
tion.* 

The  square  of  a  binomial,  the  terms  of  which  are  both  posi- 
tive, is  equal  to  the  square  of  the  Jirst  term,  +  twice  the  pro- 
duct of  the  two  terms  ;  +  the  square  of  the  last  term. 

And  the  square  of  a  residual  quantity,  is  equal  to  the 
square  of  the  first  term,  —  twice  the  product  of  the  two 
terms,  +  the  square  of  the  last  term. 


'  JL 

Ex.  1.  The  square  of  2«-f&,  is  4«2+40&- 
2.  The  square  of  A+1,  is  A2  4- 2A+1. 


For  the  method  of  finding  the  higher  powers  of  binomi- 
als, see  one  of  the  succeeding  sections. 

*  Euclid's  Elements,  Book  n.  Prop.  4. 


100  ALGEBRA. 

215.  For  many  purposes,  it  will  be  sufficient  to  express 
the  powers  of  compound  quantities  by  exponents,  without  an 
actual  multiplication. 

Thus  the  square  of  a  +  b,  is  a+6|2,or  (a+i)a  .  Art.  203. 
The  nth  power  of  bc+Q+x,  is  (&c  +  8+*)°. 
In  cases  of  this  kind,  the  vinculum  must  be  drawn  overall 
the  terms  of  which  the  compound  quantity  consists. 

216.  But  if  the  root  consists  of  several  factors,  the  vincu- 
lurn  which  is  used  in  expressing  the  power,  may  i-itlirr  ex- 
tend over  the  whole;  or  may  be  applied  to  each  of  the  fac- 
tors separately,  as  convenience  may  require. 

Thus  the  square  of  a  +  b  x  e+rf,  is  either 


-t 


or  a  +  b     Xc+d|   . 
For  the  first  of  these  expressions  is  the  square  of  the  pro- 
duct of  the  two  factors,  and  the  last  is  the  product  of  their 
squares.     But  one  of  these  is  equal  to  the  other.  (Art.  212.) 

The  cube  of  axb  +  d,  is  (oxi-f  d}3,  or  a3  x  (b  +  d}3. 

217.  When  a  quantity,  whose  power  has  been  expressed 
by  a  vinculum  and  an  index,  is  afterwards  involved  by  an  ac- 
tual multiplication  of  the  terms,  it  is  said  to  be  expanded. 

Thus  (a+b)z  ,  when  expanded,  becomes  a2  -4-2ai  +  &*  . 

And  (a-f  b  +  h)~  ,  becomes  a2  +2a6+2a/i  +  J2  +2&A  +  A2  , 

-218.  With  respect  to  the  sign  which  is  to  be  prefixed  to 
quantities  involved,  it  is  important  to  observe,  that  when  the 
root  is  positive,  all  its  powers  are  positive  also  ;  but  when  the 
root  is  negative,  the  ODP  powers  are  negative,  while  the  EVEN 
powers  are  positive. 

For  the  proof  of  this,  see  art.  109. 

The  2d  power  of  —a  is  -f-a2 
The  3d  power  is  —a3 

The  4th  power  is  +a* 

The  5th  power  is  —  a5,  &c. 

219.  Hence  any  odd  power  has  the  same  sign  as  its  root. 
But  an  even  power  is  positive,  whether  its  root  is  positive  or 
negative. 

Thus  -fax  +  a=a2 
And    —  ax—  a=a2. 

220.  A  quantity  which  is  already  a  power,  is  involved  by 
multiplying  its  index,  into  the  index  of  the  power  to  which  it  is 
to  be  raised. 

1.  The  3d  power  of  a2  ,  is  «2x3=o$ 


INVOLUTION.  101 

For  a*  —aa ;  and  the  cube  of  aa  is  aa  X  «fl  X  aa=aaaaaa=a*  ; 
which  is  the  6th  power  of  a,  but  the  3d  power  of  a8  . 
For  a  farther  illustration  of  this  rule,  see  arts.  233,  4. 

2.  The  4th  power  of  a5bz  ,  is  a3*4&2"4=a1268. 

3.  The  3d  power  of  4a2tf,  is154a**>3. 
A.  The  4th  power  x>f-2a3  x3*2d,  is  16«1! 
5.  The  5th  power  of  (a+b}*  ,  is  (a+6)10. 
<5.  The  nth  power  of  a3,  is  of". 

7.  The  7dh  power  of  (x— y)w,  is  (*•  —  y)mft 

8.  a*~+b~3~*=a*+2a3b*+b6.  (Art.  214.) 


9.  a3x63f=a«x66.         10.  (a3b* A4)3  =a96«A1*. 

221.  The  rule  is  equally  applicable  to  powers  whose  ex- 
ponents are  negative. 

Ex.  1.  The  3d  power  of  a~2,  is  «-2x3=cr*. 

Fora~2= —   (Art.  207.)     And  the  3d  power  of  this  is, 
ctct 

111        1        1 


aa  aa  aa  aaaaaa  a6 

«8 

2.  The  4th  power  of  a*  b~3,  is  a*6~!1,  or  Tfj. 

3.  The  cube  of  2xny~m,  is  8*3y-3m. 

4.  The  square  of  b3x~l,  is  b*x~z. 

5.  The  nth  power  of  .v~m,  is  #-""*,  or  ~r^. 

222.  It  must  be  observed  here,  as  in  art.  218,  that  if  the 
sign  which  is  prefixed  to  the  power  be  — ,  it  must  be  changed 
to  +>  whenever  the  index  becomes  an  even  number. 

Ex.  1.  The  square .4— a3,  is  +a6.  For  the  square  of  —  a3  ,is 
-a3  x-fl3,  which,  according  to  the  rules  for  the  signs  in  mul- 
tiplication, is  -|-a6. 

2.  But  the  cw&eof-a3,  is-«^.  For -a3  x-a3  x-a3=-ae. 

3.  The  square  of  —  xn ,  is  +xzn. 

4.  The  nth  power  of  —a3,  is  _^a3". 

Here  the  power  will  be  positive  or  negative,  according  as 
the  number  which  n  represents  is  even  or  odd. 

223.  Jl  FRACTION  is  involved,  by  involving  both  the  numera- 
tor, and  the  denominator. 


102  ALGEBRA. 

«     a* 
1.  The  square  of-^-is  ^5-    For,  by  the  rule  for  the  multi- 

plication of  fractions,   Art.  155.) 


a      a     aa     a 


J~b=bb=W  [(Art.  209.) 

2.  The  2d,  3d,  and  nth  powers  of  —  ,  arc  -r>    ~r  and  -r  • 

a         a       a3         a 


3.  The  cube  of  --,  is 


4.  The  nth  power  of  — — ,   is 

y 

5.  The  square  of  — -r 

—a~^          —a~s 

6.  The  cube  of  -£*-,  is  ~^~'    (Art.  221.) 


224.  Examples  of  binomials,  in  which  one  of  the  terms  is 
a  fraction. 

1.  Find  the  square  of  x+ £,  and  x—%,  as  in  art.  214. 


**  —\x 

-l^+i 


2  4rt     -4 

2.  The  square  of  «+~o",  is  a~  +"3" +"9" 

ft  6* 

3.  The  square  of  x+~^,  is  a;2  +6,vH--j- 

7> 

4.  The  square  of  *--,  is  *«  - 


225.  It  has  been  shown,  (Art.  165.)  that  a  co-efficient  may 
be  transferred,  from  the  numerator  to  the  denominator  of  a 
fraction,  or  from  the  denominator  to  the  numerator.  Bv  re- 


INVOLUTION.  103 

curring  to  the  scheme  of  notation  for  reciprocal  powers, 
(Art.  207.)  it  will  be  seen  that  any  factor  may  also  be  trans- 
ferred, if  the  sign  of  its  index  be  changed. 

ax~z 

1 .  Thus,  in  the  fraction  —    ,  we  may  transfer  x  from  the 

numerator  to  the  denominator. 

ax~*      a        _      a      1         a 

2.  In  the  fraction  TTj  we  may  transfer  y  from  the  de- 
nominator to  the  numerator. 

a        a      t       a  «y"~3 


da-4       d  b 

o    _  _  _  ,  A    _ 

x*   ~x3  a*  '  ayn  ~'   a 

226.  In  the  same  manner,  we  may  transfer  a  factor  which  has 
a  positive  index  in  the  numerator,  or  a  negative  index  in  the 
denominator. 

ax3        a 
1.  Thus  —  y—  .=T~^3     For  #3  is  the  reciprocal  of  AT*, 

1  3 

(Arts.  205,  207.)  that  is,  a3  =~s'     Therefore  —^—^ 


hyz  ad2      ay3 

~~  ~~ 


227.  Hence,  the  denominator  of  any  fraction  may  be  en-' 
tirely  removed,  or  the  numerator  may  be  reduced  to  a  unit, 
without  altering  the  value  of  the  expression. 

a        I 

I.  Thus       =zi,  or  «&-«. 


104  ALGEBRA. 


ADDITION  AND  SUBTRACTION  or  POWERS. 

228.  It  is  obvious  that  powers  may  be  added,  like  other 
quantitities,  by  writing  them  one  after  another,  with  their  sigHx, 
(Art.  69.) 

Thus  the  sum  of  a9  and  i*  ,  is  «*  +b*  . 

And  the  sum  of  a2  —ln  and  h'—d4,  is  a*  -ln  +h*—d4. 

229.  The  same  powers  of  the  same  letters  are  like  quantities  ; 
(Art.  45.)  and  their  co-efficients  may  be  added  or  subtracted, 
as  in  arts.  72  and  74. 

Thus  the  sum  of  2a8  and  3a8  ,  is  5a*  . 

It  is  as  evident  that  twice  the  square  of  a,  and  three  time** 
the  square  of  a,  are  five  times  the  square  of  a,  as  that  twice 
a  and  three  times  a,  are  five  times  a. 


To    —3x*y'         2bm 

Add  -2*  V         Gbm       -la*yn  6a3A*         4(a+y)n 


Sum— 53*i/5  — 4a4?/B 


/  / 


230.  But  powers  of  different  letters,  and  different  powers 
of  the  same  letter,  must  be  added  by  writing  them  down  with 
their  signs. 

The  sum  of  a*  and  a*  ,  is  a*  +a*  . 

It  is  evident  that  the  square  of  a,  and  the  cube  of  a,  are 
neither  twice  the  square  of  a,  nor  twice  the  cube  of  a. 
The  sum  of  a*  bn  and  3a'i6,  is  a3  b»  +3a'i6. 

231 .  Subtraction  of  powers  is  to  be  performed  in  the  same 
manner  as  addition,  except  that  the  signs  of  the  subtrahend 
are  to  be  changed  according  to  art.  82. 

From      2«*         -3in         3A266         a3  bn  5(a-A)s 

Sub.    +6«4          -  41"         4AU6      -aTi*^          2(a-A)6 


Diff.        8a4  -h*b*  3(a-A)« 


POWERS. 


MULTIPLICATION  OF  POWERS. 

232.  Powers  may  be  multiplied,  like  other  quantities,  by 
writing  the  factors  one  after  another,  either  with,  or  without, 
the  sign  of  inultiplieati  on  between  them.  (Art.  93.) 

Thus  the  product  of  o3  into  i%isas&2,  or  aaaltb. 

Mult,  x-3        A8J-«'        3a6jf2         dh'x-o       a*l>3y* 
Into     am          a*          —2x  <%4  a363y 

Prod.  am*-*rt*/riT'  —  6a6xy2  4^  O2i32/2a362y 


The  product  in  the  last  example,  may  be  abridged,  by 
"bringing  together  the  letters  which  are  repeated. 

It  will  then  become  asbsy^. 

The  reason  of  this  will  be  evident,  by  recurring  to  the  se- 
ries of  powers  in  art.  207,  viz. 

a*4,    a*3,     a*\     **\     a%     «-',     a"8,     «-',     «-*,  &c. 
Or,  which  is  the  same, 

1         *          *          *      *™> 

fleeter,     aaa.     aa.      a.         I,      —  >      —  >     --  >  -  ,  ou:, 

a        «a      aaa     aaaa 

By  comparing  the  several  terms  with  eacn  otner,  it  will 
be  seen  that  if  any  two  or  more  of  them  be  multiplied  to- 
gether, their  product  will  be  a  power  whose  exponent  is  the 
sum  of  the  exponents  of  the-  factors. 

Thus  a*  xa3=^aaxa#a=aaaaa=a5. 

Here  5,  the  exponent  of  the  product,  is  equal  to  2+3,  the 
aum  of  the  exponents  of  the  factors, 

So  a"  xa^raa"*1* 

For  a*  ,  is  a  taken  for  a  factor  as  many  times,  as  there  are 
units  irr  n  ; 

And  a*",  is  a  taken  for  a  factor  as  ma»y  times  as  there  are 
units  in  m  ; 

Therefore  the  product  must  be  a,  taken  for  a  factor  as  ma- 
ny times,  as  there  are  units  in  both  m  and  n.  Hence, 

233.  Powers  of  the  same  root  may  be.  multiplied,  by  adding 
ffivir  exponents. 

O 


I0«  ALGEBRA. 

Thusa»xat=as+6=o8.     And  *3  X*2  x*  = 

Mult.  4fln         3*4        fcV          «263ya          (b+h-y} 
Into     2an          2*3         Z»4y  ws62y  6  +  A  —  y 

Prod.  8a»"       £.-'/      66y4 


234.  The  rule  is  equally  applicable  to  powers  wbose  ex- 
ponents are  negative. 

ft  I         —  ^ 

L  Tims  a^2  x  a-3=a-*.     That  is  —  x  -  -=- 

an     ana      aaaaa 

11          1 

2.  ly""1*  V  ly"""1  —  u~'n~m          Tha*  is    —  V  -----  -  • 

-  y    xy    —  y  >  y"    ym~ynym 

3.  -a-2  x  a~3=  -a~s:  That  is  -  -  x  -  -  =— 

—  aa     aaa      —  .aaaaa 

I  aaa        ' 

4.  «~9xa3=a3~2=a1.  That  is  —  xa«a=  -  —a. 

aa  aa 

Indus  example,  the  exponents  are  +3,  and  —2;  and  the 
sum  of  these  is  I,  according  to  the  second  case  of  reduction 
in  addition.  (Art.  74.)  L  f^ 

I  am 

5.  a-*xam=am-n.    That  is  —  xam=-^-      • 


6.  y- 

235.  If  a+b  be  multiplied  into  a—  b,  the  product  will  be 
a~  —  bz:  (Art.  110.)  that  is, 

The  product  of  the  sum  and  difference  of  two  quantities,  is 
equal  to  the  difference  of  their  squares. 

This  is  another  instance  of  the  facility  with  which  general 
frutJis  are  demonstrated  in  algebra.  See  arts..  23  and  77. 

If  the  sum  and  difference  of  the  squares  be  multiplied), 
the  product  will  be  equal  to  the  difference  of  the  fourth 
powers,  &c. 

Thus  (a—y)  X  (a+y)=az  -f  . 


POWERS.  107 


DIVISION  OF  POWERS. 

236.  Powers  may  be  divided,  like  other  quantities,  by  re- 
^jecting  from  the  dividend  a  factor  equal  to  the  divisor  j  or 
by  placing.the  divisor  under  the  dividend,  in 'the  form  of  a 
fraction. 

Thus  the  quotient  of  «362  divided  by  bz ,  is  a3.  (Art.116.) 


Divide    9a3y4       125*  xn       a*b+ 
By       -3a3             26s           a9 

3a2y4       dx(a-h+y}* 
(a-h+y)* 

Quot.  —  3y4 

&+3y 

4                d 

** 

The  quotient  of  a5  divided  by  a3 ,  is  -5-  But  this  is  equal 
-to  a2.  For,  in  the  series 

a*4,  a+3 ,  a*2 ,  a+1,  a9,  a"1,  a~z,  «-3,  a^4,  &ic. 
fif  any  term  be  divided  by  another,  the  index  of  the  quo- 
tient will  be  equal  to  the  difference  between  the  index  of  the 
dividend,  and  that  of  the  divisor. 

aaaaa 

Thus  a*-ra3  = =a«=a«. 

aaa 

am 
So  an--ran  =—=0™-**.        Hence, 

237.  .#  power  may  le  divided  by  another  power  ofihe  same 
root,  by  subtracting  the  index  of  the  divisor  from  that  of  the 
dividend 

vyv 

Thus  f  +  f  =2T-S  =yT .      That  is  -— =y. 

y  y 

aan 
And  ontl-r  a  rsa^1-1  =  an .     That  is  — — =an . 

xn 
+xn  ?=^fl-n=cc°  =1.    That  is  -%  =1. 


108  ALGEBRA. 

Divide  jrm  ft8  8an+lw  a"**  12(6+y)« 

By        ym  b3  4om  a8  3(6 +y)3 

Quot.   y«  2?"  4(&+t,)"^ 

238.  The  rule  is  equally  applicable  to  powers  whose  ex- 
ponven.ts  are  negative. 

I .  The  quotient  of  a~*  by  a"8,  is  a~*. 

Ill         aaa        aaa        I 

That  is • v  — — .» 

.aaaaa  '  aaa      aaaa,a       1       aaaaa     aa 

-5^     -3_  -8          Th        '     — —       i          **  -1— 

3.  A3  4-A-1=A3+1=A».    That  is  A2  -4-ir=A9  xv=^3. 


In  this  example,  —  1  the  index  of  the  divisor  is  to  be  sub- 
tracted from  +2,  the  index  of  the  dividend.  But  —1  be-- 
comes  by  subtraction  -f  !•  (Art.  82.) 

4.  6an^-2a-8=3an+3.  5.  ba3  +a=ba*  . 

6.  634-i5=63-J=6-2.          7.  o*-j-ar=a-3. 
8.  (as  +y3  )m-^(a3  +*3  )"  =(a3 
9. 


„    .  . 

The  multiplication  and  division  of  powers  by  adding  and 
subtracting  their  indices,  should  be  made  very  familiar  ;  as 
they  have  numerous  and  important  applications,  in  the  high- 
er branches  of  algebra. 

EXAMPLES  OF  FRACTIONS  CONTAINING  POWERS. 

239.  In  the  section  on  fractions,  the  following  examples 
were  .omitted,  for  the  sake  of  avoiding  an  anticipation  of 
,the  subject  of  powers. 


1.  Reduce   ~-y  to  lower1  terms.      Ans.  -~—  .- 
5a4     5aaaa     5aa 


POWERS.  109 

2# 
2.  Reduce  5-77  to  lower  terras.     Ans.  -y  OF  2#. 


3.  Reduce  —  ri  -  to  lower  terms.    Ans.  —  ^~ 


8a3y— 
4.  Reduce  --  ^-;  —  --.  —  «  -  —  to  lower  terms. 


Ans.  -  3~j~2  ---  obtained  by  dividing  each  term  by  2ay. 

a*  a~3 

5.  Reduce  —  r.   and  —  zr,  to  a  common  denominator. 

o3  a 

jaz  xa"4  is  a~2,  the  first  numerator.  (Art.  146.) 
«3  X  a~3  is  a°  =1,  the  second  numerator. 
«3  x«~4  is  a"1,  the  common  denpminator. 

a-2  1 

The  fractions  reduced  ape  therefore  —  zr  and  —  pr  • 

2a*          a3 

6.  Reduce   g-y  and  —  j,  to  a  common  denominator. 

2a8         5tf*          2a3  5 

Ans,  -^  and  ^-,  or  ^  and  ^r-  (Art.  14«.) 

3*s  dx  3r7^3      3df 

7.  Multiply^-  into  ^T-  Ans-  ^  =8^  ' 

•  /  ^4 

«3+&.    «-&»     fr.y—b 

8.  Multiply  -^-,uxto  —  3- 

as-fl         J2-l 

9.  Multiply—,-,  into---- 


a 


10.  Multiply   —  ^r.into  --  , 

r  J     a,  x  y 

a*        a3  a*     i/2     a*y*      a 

11.  Divide  —  -  by  -r'       Ans.  —  x!L7=~T~T=~" 

y$  J      y2  y3  fl3  «3y3  y 

a3  —x*         x*  —  a~2 
U  Kvide  —  r-,  by  —  —  ^ 


A 


11&  ALGEBflA. 

13.  Divide  — — ,  by  *        ~- 
y  y 

14  Pivide  ^fi~:  by  —7 — •» 


. 


SECTION  IX. 


EVOLUTION  AND  RADICAL  QUANTITIES* 


A  24O1  T^  ^  quantity  is  multiplied  inter  itself,  the  pro- 
duct  is  a.  power.  On  the  contrary,  if  a  quan- 
is  resolved  into  any  number  of  equal  factors,  each  of  these 
is  a  root  o£  that  quantity. 

Thus  b  is  the  root  of  bbb ;  because  bbb  may  be  resolved 
into  the  three  equal  factors  j,  and  6,  and  b. 

In  subtraction,  a  quantity  is  resolved  into  two  parts. 

In  division,  a  quantity  is  resolved  into  two  factors. 

fa  evolution^  a  quantity  is  r-esolved  into  equal  factors. 

241.  A  root  of  a  quantity,  then,  is  a  factor  which  multiplied 
into  itself  a  certain  number  of  times  willproduce  that  quantity. 

The  number  of  times  the  root  must  be  taken  as  a  factor, 
to  produce  the  given  quantity,  is  denoted  by  the  name  of 
the  root. 

Thus  2  is  the  4th  root  of  1Q  ;  because  2x2x2x2=16, 
where  2  is  taken  four  times  as  a  factor,  to  produce  16. 

So  a3  is  the  square  root  of  a6 ;  for  a3  X  a3  =ae'.(Art.233./ 

And  a2  is  the  cube  root  of  a6-;  for  a2  Xa*  X«3=a6. 

And  a  is  the  6th  root  of  a6' ;  for  a x a  X  a x  a X a X a =a*. 

Powers  and  roots  are  correlative  terms.  If  one  quantity 
is  a  power  of  another,  the  latter  is  a  root  of  the  former. 
As63  is  the  cube  of  6;  b  is  the  cube  root  of  by'.  As  9  la- 
the square  of  3;  3  is  the  square  root,  of  9. 

242.  There  are  two  methods  in  use,  for  expressing  the 
roots  of  quantities,  one  by  means  of  the  radical  sign  V»  an(£ 
Ae  other  by  a  fractional  index.      The  latter  is  generally  to 
be  preferred.     But  the  former  has  its  uses  on  particular  oo 
easions. 

When  a  root  is  expressed  by  the  radical  sign,  the  siga 
is  placed  over  the  given  quantity,  in  this  manner  •/*• 

*  Newton's  Arithmetic,  Maclaurin,  Eoaewon,  Euler,  Saufidersoa, 

and  Simpson- 


112  ALGEBRA. 

Thus  V«  is  tiie  2d  or  square  root  of  a. 
3^/a  is  the  3d  or  cube  root, 
•ya  is  the  wth  root. 

And      Va-f-y  is  the  nth  root  of  a-fy.     • 
243.  The  figure  placed  over  the  radical  sign,  denotes  the 
number  of  factors  into  which  the  given  quantity  is  resolved;! 
in  other  words,  the  number  of  times  the  root  must  be  taken 
as  a  factor,  to  produce  the  given  quantity. 

So  that  *V«X  V«=«- 

And       V«X  V«X  V«=o. 

And       Va X  Vtt  ••••  n  times  —a. 
The  figure  for  the  square  root  is  commonly  omitted ;    •/« 
being  put  for  \Ja.       Whenever,   therefore,  the  radical  sign 
is  used  without*a  figure,  the  square  root  is  to  be  understood. 

244.  When  a  figure  or  letter  is  prefixed  to  the  radical  signr 
without  any  character  between  them,;  the  two  quantities  are 
to  be  considered  as  multiplied  together. 

,    Thus  2  •/«»  is  2  x  -vX  tnat  is>  2  multiplied  into  the  root 
of  a,  or  which  is  the  same  thing,  twice  the  root  of  a. 
And  x  V&,  is  x  X  •/&,  or  x  times  the  root  of  b. 
When  no  co-efficient  is  prefixed  to  the  radical  sign,  1  is 
always  to  be  understood ;    V«  being  the  same  as  1  ^(a,  that 
is,  once  the  root  of  a. 

245.  The  method  of  expressing  roots  by  radical  signs, 
has  no  very  apparent  connection  with  the  other  parts  of  the 
scheme  of  algebraic  notation.      But  the  plan  of  indicating 
them  by  fractional  indices,  is  derived  directly  from  the  mode 
of  expressing  poivers  by  integral  indices.     To  explain  this, 
let  a 6  be  a  given  quantity.     If  the  index  be  divided  into  any 
number  of  equal  parts,  each  of  these  will  be  the  index  of  a 
root  of  a*.  .  .„* 

Thus  the  square  root  of  a  6 ,  is  a3 .  For,  according  to  the 
definition,  (Art.  241.)  the  square  root  of  a6  is  a  factor,  which 
multiplied  into  itself  Avill  produce  a6.  But  a?.xas=atf. 
(Art.  233.}  Therefore,  a3  is  the  square  root  of  q6.  The 
index  of  me  given  quantity  «6,  is  here  divided  into  the  tw» 
equal  parts  3  and  3.  Of  course?  the  quantity  itself  is  resolr- 
ed  into  the  two  equal  factors  as  and  a3. 

The  cube  root  of  a6  is  a2.     For  a2  xa2  Xa2=os. 

Here  the  index  is  divided  into  three  equal  parts,  and  thr 
quantity  itself  resolved  into  three  equal  factors. 

The  square  root  of  rts  iso1  or  a,  For«xa=;«2. 


RADICAL  QUANTITIES.  113 

By  extending  the  same  plan  of  notation,  fractional  indi- 
ces are  obtained. 

Thus,  in  taking  the  square  root  of  a1  or  «,  the  index  1  is 

divided  into  the  two  equal  parts  *  and  -|;  and  the  root  is  a** 
On  the  same  princple, 

The  cube  root  of  o,  is  a3  =  3-\/a 
The  fourth  root  is  a7  =  '*•/« 
The  nth  root,  is  a?  —  Va,  &c- 

L       „   . . 

And  the  nth  root  of  a+x,  is  (a+^)n  =  va+x. 

246.  In  all  these  cases,  the  denominator  of  the  fractional 
index,  expresses  the  number  of  factors  into  which  the  given 
quantity  is  resolved. 

So  that  a?x  a?=a. 
i.         i         JL 
a3  X«7xa3=a. 

a*  xa"....  n  times  =d.  See  art.  243. 

247.  It  follows  from  this  plan  of  notation,  that 

a¥X  a¥  =  a^"*"¥.  Fora*     ^=a*  or  a. 

0s  x  a7  X  a^=e:  a3~~'~7~'~7=a1   &c. 

where  the  multiplication  is  performed  in  the  same  manner, 
as  the  multiplication  of  powers,  (Art.  233,)  that  is,  by  ad- 
ding the  indices. 

248.  Every  root  as  well  as  every  power  of  1  is  1.     (Art. 
209.)     For  a  root  is  a  factor  which  multiplied  into  itself  will 
produce  the  given  quantity.     But  no  factor  except  1  can 
produce  1,  by  being  multiplied  into  itself. 

So  that  1B,  1,  yi,  VI,  &c-  are  all  equal. 

249.  Negative  indices  are  used  in  the  notation  of  roots,  as 
well  as  of  powers.     See  art.  207. 

Thus    —=a~"*  -j=a""* 

a*  a* 


114  ALGEBRA. 


POWERS  OF  ROOTS. 

250.  It  has  been  shewn  in  what  manner  any  power  or  root 
may  be  expressed  by  means  of  an  index.  The  index  »f  a 
•power  is  a  whole  number.  That  of  a  root  is  a  fraction 
whose  numerator  is  1.  There  is  also  another  class  of  quan- 
tities, which  may  be  considered,  either  as  powers  of  roots, 
or  roots  of  powers. 

Suppose  a*  is  multiplied  into  itself,  so  as  to  be  repeated 
•three  times  as  a  factor. 


The  product  a-2"1"     r~*  or  a*  (Art.  2-17.)  is  evidently  the 

i 

cube  of  a*,  that  is,  the  cube  of  the  square  root  of  «.  This 
fractional  index  denotes,  therefore,  a  power  of  a  root.  The 
denominator  expresses  the  root,  and  the  numerator  the  pow- 
er. The  denominator  shows  into  how  many  equal  factors  or 
f  roots  the  given  quantity  is  resolved ;  and  the  numerator 
shows  how  many  of  these  roots  are  to  be  multiplied  to- 
gether. 

Thus  a7  is  the  4th  power  of  the  cube  root  of  a. 

The  denominator  shows  that  a  is  resolved  into  the  three 

111. 

factors  or  roots  a7,  and  a7,  and  a5.  And  the  numerator 
shows  that  four  of  these  are  to  be  multiplied  together;  which 

will  produce  the  fourth  power  of  a7  ;  that  is, 

11114 

a7  X  a7  X  a7  X  a7  =  a7. 

251.  As  a*  is  a  power  of  a  root,  so  it  is  a  root  of  a  power. 
Let  a  be  raised  to  the  third  power  a*.     The  square  root  of 

this  is  aT.  For  the  root  of  as  is  a  quantity  which  multipli- 
ed into  itself  will  produce  a3  . 

But  according  to  art.  247,  «?  =  fl¥  X  a*  X  a¥  ;    and  tjlis 
multiplied  into  itself  (Art.  103.)  is 

i        i        i        i        i        i 
a?  x  a?  X  a8  x  a*  X  aT  X  a2  =a*  . 

3  ^ 

Therefore  a?  is  the  square  root  of  the  cube  of  a. 

m 

In  the  same  manner,  it  may  be  shown  that  a"   is   the    wtb 
power  of  the  nth  root  of  a;  or  the  nth  root  of  the  mth  pow^ 


RADICAL  QUANTITIES.  115 

<r..:|JL' 

«r:  that  is,  a  root  of  a  power  is  equal  to  the  same  power  of  the 
same  root.  For  instance,  the  fourth  power  of  the  cube  root 
of  a,  is  the  same,  as  the  cube  root  of  the  fourth  power  of  a. 

252.  Roots,  as  well  as  powers,  of  the  same  letter  may  be 
multiplied  by  adding  their  exponents.  (Art.  247.)     It  will  be 
easy  to  see,  that  the  same  principle  may  be  extended  to 
powers  of  roots,  when  the  exponents  have  a  common  de- 
nominator. 

Thus  a*x«*  =  a*  +  T=0*. 

For  the  first  numej-ator  shows  how  often  aT  is  taken  as  a 

* 
factor  to  produce  «T.     (Art.  250.) 

And  the  second  numerator  shows  how  often  «T  is  taken  as 

s 
a  factor  to  produce  «T. 

The  sum  of  the  numerators,  therefore,  shows  how  often 
the  root  must  be  taken,  for  the  product.  (Art.  103.) 

i.       i        ! 
Or  thus,  a7  =  aTX  OT. 

3111 

And        aT  =  aT  x  aT  X  a7. 

SSlllltff 

Therefore  ay  x  aT  =  aT  x  aT  X  aT  X  aT  x  «T  =  aT. 

253.  The  value  of  a  quantity  is  not  altered,  by  applying 
to  it  a  fractional  index  whose  numerator  and  denominator 
are  equal. 

t     "      3          n 

Thus  «  =  ef2'  =  a"^  =  an.  For  the  denominator  shows 
that  a  is  resolved  into  a  certain  number  of  factors;  and  the 

n 

numerator  shows  that  all  these  factors  are  included  in  a" . 

-i          -1  JL 

Thus  a*  =  a?  x  a2,  which  is  equal  to  a.  (Art.  246.) 

Sill  • 

And    aT=  aT  x  aT  X  a7,  which  is  also  equal  to  a. 
-   «        i        i        i 

And    a»  =  a"  x  a"  X  a" . . . .  n  times. 

On  the  other  hand,  when  the  numerator  of  a  fractional 
index  becomes  equal  to  the  denominator,  the  expression 
may  be  rendered  more  simple  by  rejecting  the  index. 

n 

Instead  of  a"  ,  we  may  write  a. 

254.  The  index  of  a  power  or  root  may  be  exchanged, 
for  any  other  index  of  the  same  value. 

2  4 

Instead  of  aT,  we  may  put  «6. 


116  ALGEBRA. 

For,  in  the  latter  of  these  expressions,  a  is  supposed  to  be 
resolved  into  twice  as  many  factors  as  in  the  former ;  and  the 
numerator  shews  that  twice  as  many  of  these  factors  are  to 
be  multiplied  together.  So  that  the  whole  value  is  not  al- 
tered. 

i         i         s 

The  one  is     «7  x  a7  =  a7. 

The  other  is  a*  x  «T  X  a®  X  a*  =  a*. 

2  3n 

On  the  same  principle  a7=«7»». 


'a?7=,rT  =  ,r"5',  &c.  that  is,  the  square  of  the  cube 
root  is  the  same,  as  the  fourth  power  of  the  sixth  root,  the 
sixth  power  of  the  9th  root,  &c. 

4  6  an 

So  a*=a*=a7=  a~^.    For  the  value  of  each  of  these 
indices  is  2.  (Art.  135.) 

255.  From  the  preceding  article,  it  will  be  easily  seen,  that 
a  fractional  index  may  be  expressed  in  decimals. 
JL       _* 

1 .  Thus  aa  =  olirora0>s;  that  is,  the  square  root  is  equal 

to  the  5th  power  of  the  tenth  root, 
i        _?  5_  . 

2.  a4  =  a  1T°  or  a-0-2  *  ;  that  is,  the  fourth  root  is  equal  to- 
the  25th  power  of  the  100th  root. 

3.  a*=a0-4.  5.    a^a1-*. 

4,  aT=a3'5.  G.    a  *  =«2'75. 

In  many  cases  however  the  decimal  can  be  only  an  ap- 
proximation to  the  true  index. 

Thus  a7=a°'3  nearly. 

a7=a°*33  more  nearly. 

a7  =a°  • 3  3  3  3  3  very  nearly. 

In  this  manner,  the  approximation  may  be  carried  to  any 
degree  of  exactness  which  is  required. 

Thus  a*=al'666&s. 

5 
aT==ae.8  3  3  3  3^ 

These  decimal  indices  form  a  very  important  class  of  nura- 
fcers,  caUed  logarithms. 


EVOLUTION.  117 

It  is  frequently  convenient  to  vary  the  notation  of  powers 
of  roots,  by  making  use  of  a  vinculum,  or  the  radical;  sign  •/• 
In  doing  this,  we  must  keep  in  mind,  that  the  power  of  a  root 
is  the  same,  as  the  root  of  a  power  ;  (Art.  251.)  and  also,  that 
the  denominator  of  a  fractional  exponent  expresses  a  root, 
and  the  numerator,  a.  power.  (Art.  250.) 

2  1\.  1 

Instead,  therefore,  of  aT,  we  may  write  (rt7'2,  or  («2)7,  or 
V«2- 

The  first  of  these  three  forms,  denotes  the  square  of  the 
cube  root  of  a  ;  and  each  of  the  two  last,  the  cube  root  of 
the  square  of  a. 

*    ~Tim  ~^     - 

So  a*    =  a    j    =  a  |    =  V«"'- 

3 

And     bx*  =  b3x3*  = 


And  a 


EVOLUTION. 

257.  Evolution  is  the  opposite  of  involution.  One  is 
finding  a  power  of  a  quantity,  by  multiplying  it  into  itself. 
The  other  is  finding  a  root,  by  resolving  a  quantity  into 
equal  factors.  A  quantity  is  resolved*  into  any  number  of 
equal  factors,  by  dividing  its  index  into  as  many  equal  parts. 
(Art.  245.) 

Evolution  may  be  performed,  then,  by  the  following  gene- 
ral rule; 

Divide  the  index  of  the  quantity,  by  the  number  expressing 
the  root  to  be  found. 

Or,  place  over  the  quantity  the  radical  sign  belonging  to 
the  required  root. 

1.  Thus  the  cube  root  of  ft6  is  a3.    For  a2  xa3  Xa2  —  o6. 
Here  6,  the  index  of  the  given  quantity,  is  divided  by  3, 

the  number  expressing  the  cube  root. 

2.  The  cube  root  of  a  or  a1,  is  a7  or  s\/a. 

For  «T  x  «T  X  a7,  orVa  X  V«  X  V«=«.  (Arts.  243, 246.) 
Here  the  index  1  is  divided  by  3. 

3.  The  5th  root  of  ab,  is  (a£)~*  or 


115  ALGEBRA. 


_ 

4.  The  nth  root  of  a*,  is  n"   or  Va*. 

5.  The  7th  root  of  2d-c>  is  (2d-x}^  or  7v/2</^-*:. 


6.  The  5lh  root  of  a  —x\  ,  is  a"^-~HTor  * 

7.  The  cube  root  of  a*,  is  a*.    (Art.  103.) 

8.  The  4th  root  of  a"1,  is  a~*. 

9.  The  cube  root  of  tt*,  is  a^. 

10.  The  nth  root  of  xm,  is  a;^. 

258.  According  to  the  rule  just  given,  the  cube  root  of 
the  square  root  is  found,  by  dividing  the  index  j-  by  3,  as  in 
example  7th.  But  instead  of  dividing  by  3,  we  may  multi- 
ply by  $.  Forf*T3  =  i*4=rixi.  (Art.  1G2.) 

1  1       1 

So~--^n=—  x  -•    Therefore  the  with  root  of  the  nth 

/fv  if  I  t\t 

-XL 
root  of  a  is  equal  to  an      '"• 

L 

~T\m     icXL       L 

rpi      i  •         ^'     —  «'N«i          nm 

That  is,  an\    •  =a 

Here  the   two  fractional  indices  are  reduced  to  one  by 
multiplication.  » 

•*        It  is  sometimes  necessary  to  reverse  this  process;  to  re- 
solve an  index  into  two  factors. 

Thus^=^X^  =  ^|  .     That  is,  the  Sth  root  of  *  is 
equal  to  the  square  root  of  the  4th  root. 

•*?      .  i 

_  L  _i  y  JL  HII" 

Prt  i    7  \mn  i     L  m        n  i     1.1*1 


It  may  be  necessary  to  observe,  that  resolving  the  index 
into  factors,  is  not  the  same  as  resolving  the  quantity  into  fac- 
tors. The  latter  is  effected,  by  dividing  the  index  into  parts. 

259.  The  rule  in  art.  257,  may  be  applied  to  every  case 
in  evolution.  But  when  the  quantity  whose  root  is  to  be 
found;  is  composed  of  several  factors,  there  will  frequently  be 
an  advantage  in  taking  the  root  of  each  of  the  factors  sepa- 
rately. 

This  is  done  upon  the  principle,  that  the  root  of  the  pro- 
duct of  several  factors  i  is  equal  to  the  product  of  their  roots. 


EVOLUTION.  119 

Thus  Vab  =  Vax  -Jb.  For  each  member  of  the  equation, 
if  involved,  will  give  the  same  power. 

The  square  of  V~ab  is  ab.    (Art.  241.) 
The  square  of  Jax^b,  is  V«  X  <Ja  X  ^b  X  Jb.  (Art.  102.) 
But  t/ax-v/a^a.  (Art.  241.)    And  V&XA/&=&. 
Therefore  the  square 
which  is  also,  the  square  of 

L         i    £ 

On  the  same  principle,  (ab)n  —anbn. 

When,  therefore,  a  quantity  consists  of  several  factors,  we 
may  either  extract  the  root  of  the  whole  together  ;  or  we 
may  find  the  root  of  the  factors  separately,  and  then  multi- 
ply them  into  each  other. 

i  i    i 

Ex.  1.   The  cube  root  of  xy,  is  either  (xy}^,  or  x*y^. 

2.  The  5th  root  of  3y,  is  V§y  or  s  -/3x  Vy. 


3.  The  6th  root  of  abh,  is  (abh}*,  or 

4.  The  cube  root  of  Sb,  is  (86)T,  or  26T. 

5.  The  nth  root  of  xny,  is  (x  y)n  ,  or  xy  ". 

260.   The  root  of  a  fraction  is  equal  to  the  root  of  the  nu- 
merator divided  by  the  root  of  the  denominator. 


_ 

1.  Thus  the  square  root  of  ~r  --  1. 

^ 

i  L         L 

a     a"          an     an  a 

2.  Sothenthrootof;-=-7.  For-rx—  •  •  •»  times=T' 

6"  b~n     6« 

x         Jx 

3.  The  square  root  of  —  ,  is  -r—  . 


261.  For  determining  what  «;§•»  to  prefix  to  a  root,  it  is 
important  to  observe,  that 


120  ALGEBRA. 

Jin  odd  root  of  any  quantity  has  the  same  sign  as  the  quan- 
tity itself; 

An  even  root  of  an  affirmative  quantity  is  ambiguous  ; 

Jin  even  root  of  a  negative  qua  at  in/  is  impossible. 

That  tlie  3d,  5th,  7th,  or  any  other  mid  root  of  a  quanti- 
ty, must  have  the  same  sign  as  the  quantity  itself,  is  evident 
from  art.  219. 

262.  But  an  even  root  of  an  affirmative  quantity,  may  be 
cither  affirmative  or  negative.    For  the  quantity  may  be  pro- 
duced from  the  one,  as  well  as  from  the  other.  (Art.  219.) 

Thus  the  square  root  of  a2  is  +a  OP  — a. 
An  even  root  of  an  affirmative  quantity  is,  therefore,  said 
to  be  ambiguous,  and  is  marked  with  both  -}-  and  — . 

Thus  the  square  root  of  3b,  is  _   &*• 

The  4th  root  of  a?,  is  _x*. 

263.  But  no  even  root  of  a  negative  quantity  can  be  found. 
Thus  the  square  root  ot*a*  is  neither  -fa  nor  —a. 

For -fax +a=  +  a3.         And  —ax  —a  =  -fa3  also. 

An  even  root  of  a  negative  quantity  is,  therefore,  said  to 
be  impossible  or  imaginary. 

There  are  purposes  to  be  answered,  however,  by  applying 
the  radical  sign  to  negative  quantities.*  The  expression  V  —  a 
is  often  to  be  found  in  algebraic  processes.  For,  although 
we  are  unable  to  assign  it  a  rank,  among  either  positive  or 
negative  quantities ;  yet  we  know  that  when  multiplied  into 

itself,  its  product  is  —a,  because  ^  —  a  is  by  notation  a  root 
of  —  a,  that  is,  a  quantity  which  multiplied  into  itself  produ- 
ces —  a. 

This  may,  at  first  view,  seem  to  be  an  exception  to  the 
general  rule  that  the  product  of  two  negatives  is  affirmative. 
But  it  is  to  be  considered,  that  V— a  is  not  itself  a  negative 
quantity,  but  the  root  of  a  negative  quantity. 

It  ought  also  to  be  observed  that  V— a  is  not  equivalent 
to  —  V«-  The  first  is  a  root  of  —a,  but  the  latter  is  a  root 
of+o. 

For—  V«X  —  ^/a  —  +  a. 

*  See  an  interesting  Essay,  on  the  use  of  impossible  quantities  in 
calculation,  by  Professor  Playfair,  in  the  London  Philosophical  Trans- 
actions, for  1778. 


EVOLUTION.  121 

.  'The  methods  of  extracting  the  roots  of  compound 
Quantities  are  to  be  considered  in  a  future  section.  But 
there  is  one  class  of  these,  the  squares  of  binomial  and  resid- 
ual quantities,  which  it  will  be  proper  to  attend  to  in  this 
place.  It  has  been  shown,  (Art.  214.)  that  the  square  of  a 
binomial  quantity  consists  of  three  terms,  two  of  which  are 
complete  powers,  and  the  other  is  a  double  product  of  the 
joots  of  these  powers.  The  square  of  a  +  6,  for  instance,  is 


two  terms  of  which,  a*  and  62,  are  complete  powers,  and 
2ab  is  twice  the  product  of  a  into  b,  that  is,  of  the  root  of  a* 
into  the  root  of  Ist. 

Whenever,  therefore,  we  meet  with  a  quantity  of  this  de- 
scription, we  may  know  that  its  square  root  is  a  binomial  ; 
and  this  may  be  found,  by  taking  the  root  of  the  two  terms 
which  are  complete  powers,  and  connecting  them  by  the 
sign  +  .  The  other  term  disappears  in  the  root.  Thus  ta 
find  the  square  root  of 


take  the  root  of  #*,  and  the  root  of  y*,  and  connect  then* 
by  the  sign  +.  The  binomial  root  will  then  be  x+y. 

In  a  residual  quantity,  the  double  product  has  the  sign  — 
prefixed,  instead  of  +•  The  square  of  a  —  6,  for  instance, 
is  a2  '  —  2a6  +  i2.  (Art.  214..)  And  to  obtain  the  root  of  a 
quantity  of  this  description,  we'  have  only"  to  take  the  roots 
of  the  two  complete  powers,  and  connect  them  by  the  sign' 
—  .  Thus  the  square  root  of  'xz  —Zzy+y*  is  x—  y.  Hence, 

265.  To  extract  a  binomial  or  residual  square  root,  take 
the  roots  of  the  two  terms  u-hich  are  complete  powers,  and  con- 
nect them  by  the  sign,  which  is  prefixed  to  the  other  term. 

Ex.  1.     Find  the  root  of  x*  +2^+1. 

The  two  terms  which  are  complete  powers  are**'8  and  1* 

Their  roots  are  x  and  1.     (Art.  248.) 

The  binomial  root  is,  therefore,  .r+L 

2.  The  square  root  of  x2  —2^+1,  is  x  —  1.     (Art.  214.) 

3.  The  square  root  of  az  +  a+|,  is  a+|.     (Art.  224.) 

4.  The  square  root  of  a3+|  «  +  £,  is  o+f. 

b*  I 

5.  The  square  root  of  a2  +  a5+"T"}  is  a+-^' 


b 

root  of  «3-f +~r,  is  a+ 

e       c85 

R 


6.  The  square  root  of  «2  +  ~~+~Tj  Is  ct+—' 

v  C  C 


ALGEBRA* 

2G6.  A  root  whose  value  cannot  be  exactly  expressed  in' 
numbers,  is  called  a  SURD. 

Thus  -v/2  is  a  surd,  because  the  square  root  of  2  cannot 
be  expressed  in  numbers,  with  perfect  exactness. 

In  decimals,  it  is  1.41421356  nearly. 

But  though  we  are  unable  to  assign  the  value  of  such  a 
quantity  when  taken  alone,  yet  by  multiplying  it  into  itself,  or 
by  combining  it  with  other  quantities,  we  may  produce  ex- 
pressions whose  value  can  be  determined.  There  is  there- 
fore a  system  of  rules  generally  appropriated  to  surds.  But 
as  all  quantities  whatever,  when  under  the  same  radical  sign, 
or  having  the  same  index,  may  be  treated  in  nearly  the  same 
manner;  it  will  be  most  convenient  to  consider  them  togeth- 
er, under  the  general  name  of  Radical  Quantities  ;  under- 
standing by  this  term,  every  quantity  which  is  found  under  a 
radical  sign,  or  which  has  a  fractional  index. 

267.  Every  quantity  which  is  not  a  surd,  is  said  to  be  ra- 
tional. But  for  the  purpose  of  distinguishing  between  radi- 
cals and  other  quantities,  the  term  rational  will  be  applied,  in 
this  section,  to  those  only  which  do  not  appear  under  a,  radi- 
cal sign,  and  which  have  not  a  fractional  index. 

/r 

REDUCTION  OF  RADICAL  QUANTITIES. 

J268.  Before  entering  on  the  consideration  of  the  rule*  for 
the  addition,  subtraction,  multiplication,  and  division  of  rad- 
ical quantities,  it  will  be  necessary  to  attend  to  the  method* 
«f  reducing  them  from  one  form  to  another. 

First,  to  reduce  a.  rational  quantity  to  the  form  of  a  radical; 

Raise  tfie  quantity  to  a  power  of  the  same  name  as  the  given 
root,  and  then  apply  the  corresponding  radical  sign  or  index. 

Ex.  1.     Reduce  a  to  the  forrn  of  the  rath  root. 

The  nth  power  of  a  is  a*.     (Art.  211.) 

Over  this  place  the  radical  sign,  and  it  becomes  *Va". 

It  is  thus  reduced  to  the  form  of  a  radical  quantity,  with- 

•  i 

out  any  alteration  of  its  value.     For  Va"  =a"=a.       (Art, 
253.) 

3L  Reduce  4  to  the  form  of  the  cube  root. 
Ans.   V64  or 


HADICAL  QUANTITIES.  123 

3.  Reduce  3a  to  the  form  of  the  4th,root. 

Ans.  Vela7. 

4.  Reduce  }al  to  the  form  of  the  square  root. 

Ans.       as^i 


5.  Reduce  3  X  a— x  to  the  form  of  the  cube  root. 

Ans.    ^27xa^x\3.     See  art.  213. 

6.  Reduce  aa  to  the  form  of  the  cube  root. 
'The  cube  of  a*  is  a-6 .  (Art.  22CL) 

—"I 

And  the  cube  root  of  a6  is  8-v/a6  =a6j7. 

In  cases  of  this  kind,  where  a  power  is  to  be  reduced  to 
the  form  of  the  nth  root,  it  must  be  raised  to  the  nth  power, 
not  of  the  given  letter,  but  of  the,  power  of  the  letter. 

Thus  in  the  example,  a6  is  the  cube,  not  of  a,  but  of  a*. 

7.  Reduce  a364  to  the  form  of  the  square  root. 

Ans.  vV&8,  or  (a6b8}*. 

8.  Reduce  am  to  the  form  of  the  nth  root. 

Ans.  an'V . 

269.  Secondly,  to  reduce  quantities  which  have  different 
indices,  to  others  of  the  same  value  having  a  common  index  ; 

1.  Reduce  the  indices  to  a  common  denominator; 

2.  Involve  each  quantity,  to  the  power  expressed  by  the 
numerator  of  its  reduced  index. 

3.  Take  the  root  denoted  by  (he  common  denominator. 

i  '? 

Ex.  1.  Reduce  «*  and  Is  to  a  common  index, 

1st.  The  indices  £  and  £  reduced  to  a  common  denomi- 
nator, are  T3¥  and  T\.  (Art.  146.) 

2d.  The  quantities  .a  and  b  involved  to  the  powers  expres- 
sed by  the  two  numerators,  are  «3  and  b2. 

3d.  The  root  denoted  by  the  common  denominator  is  T^. 

The  answer,  then,  is  a3  T^  and  bz  T^. 

The  two  quantities  are  thus  reduced  to  a  common  index, 
^without  any  alteration  in  their  values. 

For  by  art.  254,  a:i=  «1¥,  which  by  art.  258,  =^«»p*. 

J  tn 1 

And  universally  aft  =  a'nm  ™  «m|^c. 


ALGEBRA. 

1  2 

2.  Reduce  aj  and  bx*  to  a  common  index. 

The  indices  reduced  to  a  common   denominator  are  |- 
and  £. 

3  4  ___  

"JThc  quantities,  then,  are  a*  and   (kv)«,  or  a* 

i.  i          JL 

3.  Reduce  a2  ancU" .     Ans.   «2nj"and6n. 

J.  J.  JL  _ 

4.  Reduces"    and  y'".   Ans.   Arm\mnand  j/"j> 

5.  Reduce 2*  and  3*.     Ans.  8*  and  9*. 


1 


1 

6.  Reduce  (a -J-J)2  and  (x— y}"*. Ans. a+6  j  andx— y     ' 

270.  When  it  is  required  to  reduce  a  quantity  to  a  given 
index ; 

Divide  the  index  of  the  quantity  by  the  given  index,  place 
the  quotient  over  the  quantity,  and  set  the  given  index  over 
the  whole. 

This  is  merely  resolving  the  original  index  jnto  two  fac- 
tors, according  to  art.  258. 

i 
]Ex.  1.  Reduce  af  to  the  index  i. 


This  is  the  index  to  be  placed  over  a,  which  then  becomes 

—  -  1 

j 

.a?  ;  and  the  given  index  set  over  this  makes  it  a3    ,  the  aiv- 


ewer. 

8. 

2.  Reduce  a2  and  x*,  to  the  common  index  J-. 
24-  1  =2x3=6,  the  first  index          ) 

-2-7-.i  =  fx3=!-,  the  second  index     5 

•    i  »i 

Therefore  (a6)3  aqd  (x-"2)7  are  the  quantities  required. 

i  JL 

3.  Reduce  4"2   and  33,to  the  common  index  £. 
Answer.     (43')^"  and  (32)'«\ 

271.  Thirdly,  to  remove  a  part  of  a  root  from  under  the 
Radical  sign  ; 

If  the  quantity  can  be  resolved  into  two  factors,  one  of 
which  is  an  exact  power  of  the  same  name  \vith  the 
;-oot;  find  the  root  of  this  power,  and  prefix  it  to  the  other  fac- 
tor, with  the  radical  sign  between  them. 


RADICAL  QUANTITIES.  125 

This  rule  is  founded  on  the  principle,  that  the  root  of  the 
product  of  two  factors  is  equal  to  the  product  of  their  roots. 
(Art.  259.) 

It  will  generally  be  best  to  resolve  the  radical  quantity  into 
such  factors,  (hat  one  of  them  shall  be  the  greatest  power 
which  will  divide  the  quantity  without  a  remainder.  If  there 
is  no  exact  power  which  will  divide  the  quantity,  the  reduc- 
tion can  not  be  made. 

Ex.  1.  Remove  a  factor  from  -\/S. 

The  greatest  square  which  will  divide  8  is  4. 

We  may  then  resolve  8  into  the  factors  4  and  2.  For  4x2=8. 

The  root  of  this  product  is  equal  to  the  product  of  the 
roots  of  its  factors;  that  is,  y'8=^4xv'2. 

But  ^4=2.  Jnstea(l  of  ^/4}  therefore,  we  may  substitute 
its  equal  2.  We  then  have  2  x  v^2  or  2-/2. 

This  is  commonly  called  reducing  a  radical  quantity  to  its 
•most  simple  terms.  But  the  learner  may  not  perhaps  at  once 
perceive,  that  2^2  is  a  more  simple  expression  than  -v/8. 

2.  Reduce    Va2x.     Ans.    -/a2  x^x=cax  ^x=a^/x. 

3.  Reduce    VlS.     Ans.  v/9~x2=v/9x  ^2= 

4.  Reduce    3  JQ&b*c.     Ans.  3  V'6461  x  V^  = 


5.  Reduce   *J—.  Ans.  -4J—.     (Art.  260.) 

V  c*d  c    v  cd 

6.  Reduce     vd*p.     Ans.  a  n^/b,  or  aln. 

7.  Reduce  (as-azby*.      Ans.  a(a-b)?. 

8.  Reduce    54a6&^.     Ans. 


272.  By  a  contrary  process  the  co-efficient  of  a  rad;cal 
quantity  may  be  introduced  under  the  radical  sign. 


For  a=  V«n  o%  fi«-  (Art.  253.)  And  V«n  X  V&=  V«  &• 
Here  the  co-dficient  a  is  first  raised  to  a  power  of  the 
same  name  as  the  radical  part,  and  is  then  introduced  as  a 
factor  under  the  radical  sign. 


1 
2.     a(x— I]*  =  (a3 


12T>  ALGEBRA. 


ADDITION  AND  SUBTRACTION  OF  RADICAL  QUANTITIES 

273.  Radical  quantities  may  be  added  like  rational  quan- 
tities, by  writing  tkem,  one  after  another  with  their  signs.    (Art. 


Thus  the  sum  of   \'a  and  /&,  is 

li  1  1  i  1  1  -L 

And  the  sum  of  a*  —  A7  and  a?T  —  yn  ,  is  «"*  —  A7  +  x*  —  i/n 
But  in  many  cases,  several  terms  may  be  reduced  to  one« 
as  in  arts.  72  and  74. 

The  sum  of  2  /a  and  3  /a  is  2/a  +  3/o=5/o. 

For  it  is  evident  that  twice  the  root  of  a,  and  three  times 
the  root  of  a,  are  five  times  the  root  of  a.  Hence, 

274.  When  the  quantities  to  be  added  have  the  same  rad- 
ical part,  under  the  same  radical  sign  or  index  ;  00*0*  the  ra- 
tional parts,  and  to  the  sum  annex  -the  RADICAL  PARTS. 

If  no  rational  quantity  is  prefixed  to  the  radical  sign,  1  is 
always  to  be  understood.  (Art.  244.) 


To 

Add 


275.  If  the  radical  parts  are  originally  different,  they  may 
sometimes  be  made  alike,  by  the  reductions  in  the  preceding 
articles. 

1.  Add  /8  to  /50.      Here  the  radical  parts  are  not  the 
same.     But  by  the  reduction  in  art.  271,    /8=2/2,  and 

V50=5/2.     The  sum  then  is  7/2. 

2.  Add  /1G6  to  /46.         Ans.  4/6  +  2/6=6/6. 

3.  Add  /a2 x  to  Jb*> 


4.  Add    (S6k*»y)     to  (25y),    Ans. 
».  Add  VI 80  to  3/2o. 


RADICAL  QUANTITIES. 

276.  But  if  the  radical  parts,  after  reduction,  are  different, 
or  have  different  exponents,  they  cannot  be  united'  in  the 
same  term  ;  and  must  be  added  by  writing  them  one  after 
the  other. 

The  sum  of  3^/1  and  2Ja,  is  3Vb+2^/a. 

It  is  manifest  that  three  times  the  root  of  6,  and  twice  the 
root  of  a,  are  neither  five  times  the  root  of  5,  nor  five  times 
the  root  of  a,  unless  b  and  a  are  equal. 

The  sum  of  *  i/a  and  3  -/«,  is  2  V«+  3  \/«. 
The  square  root  of  a,  and  the  cube  root  of  a,  are  neither 
twice  the  square  root,  nor  twice  the  cube  root  of  a. 

277.  Subtraction  of  radical  quantities  is  to  be  performed 
in  the  same  manner  as  addition,  except  that  the  signs  in  the 
subtrahend  are  to  be  changed  according  to  art.  82. 


From     Jay      4 
-3 


' 


From  ^50,  subtract  JQ..  Ans.  5/2—  2v/2=3v/2.(Art.275.) 
From  S^b4y,  subtract  sV&y4.-    Ans.  (b—  y)x  3\/%- 
From  V^j  subtract  '\/x. 

MULTIPLICATION  OF  RADICAL  QUANTITIES. 

Radical  quantities  may  be  multiplied,  like  other  quanti- 
fies, by  writing  the  factors  one  after  another,  either  with  or 
without  the  sign  of  multiplication  between  them.  (Art.  93.)i 

Thus  the  product  of  ^a  into  \fb,  is 


The  product  of  AT  into  y$  is 

But  it  is  often  expedient  to  bring  the  factors  under  the 
same  radical  sign.  This  may  be  done,  if  they  are  first  re- 
duced to  a  common  index. 

Thus  nJxxnVy—nVxy-  For  the  rqot  of  the  product  of 
several  factors  is  equal  to  the  product  of  their  roots.  (Art. 
259.)  Hence, 

£79.  Quantities  under  the  same  radical  sign  or  index, 


123  ALGEBRA. 

lie  multiplied  together  like  rational  quantities,  the  product  being 
placed  under  the  common  radical  sign  or  index.* 

i   ^          i 
Multiply  '•/r  into  3^/y,  that  is,  x*  into  y7. 

The  quantities  reduced  to  the  same  index,  (Art.  269.)  are 

i    '  i  »     6    

(#3)T,  and  (y2)*,  and  their  product  is  (#3y8)8  =  vx3y*. 

3  i  JL 

Mult,     vo  +  m     y/dx         a3         (a+y}n  a"' 


Into       Va  —  m     >%          x*         (b+hf 


A" 


Prod,    v/a2— »i*  (a3x}*  (anxm}m* 


Multiply    ^8xb  into    V2xb.     Prod.  VlG^2  =4*&. 
In  this  manner  the  product  of  radical  quantities  often  be- 
comes rational. 

Thus  the  product  of  -v/2  into  v/18=-/36=6. 

1.  1  i 

And  the  product  of  (a37/3)4into(a2y)4=  («*i/4)4=ay. 

280.  Roots  of  the  same  letter  or  quantity  may  be  multiplied, 
ly  adding  their  fractional  exponents. 

The  exponents,  like  all  other  fractions,  must  Be  reduced 
to  a  common  denominator,  before  they  can  be  united  in  one 
term.  (Art.  148.) 

1  L  I     I     1  '•       3    I    2  5 

Thus  a^x  a3  =  a^T"J=aTT  <r=  aT. 
The  values  of  the  roots  are  not  altered,  by  reducing  their 
indices  to  a  common  denominator.  (Art.  254.) 

-  - } 
Therefore  the  first  factor  a2  =  a6  f 

-  2  t 
And  the  second                 a3  =  a6  ) 

s        11        i 
But    a*=saTxa¥xa  •  (Art.  250.) 

3  i  L  * 

And    a*=  a6  x  a6.  [103,  250  > 

1  I.  J.  i  i  5 

The  product  therefore  isaex  oaxa°x  afixa6  =  a^.(Art. 
And  in  all  instances  of  this  nature,  the  common  denomi- 
nator of  the  indices  denotes  a  certain  root,  and  the  sum  of 

*The  case  of  an  imaginary  root  of  a  negative  quantity  is  an  excep- 
tion. (Art.  263.) 


RADICAL  QUANTITIES. 

fhe  numerators  shows  how  often  this  is  to  be  repeated  as  at 
Jiictor  to  produce  the  required  product* 

3.  J_  ny  n  m+n 

Thus  a"  X  «m=  a»»»  X  a»»  =  a»'». 

?,fult.  ~~ 

Into' 

Prod. 


1  i  8  2  1 

The  product  of  y*  into  y     3   »JT     6  =yr • 
Here  the  sum  of  the  indices  |  and  - 1  is  f ,  according  to 
Uie  rule  for  reduction  in  addition.  (Art.  74,  or  148.) 


The  product  of  a"  into  a    n,isan     n  =a° —I, 


. 
The  product  of  a2  into  a^=  aT  x  «3  =  «3. 

281.  From  the  last  example,  it  will  be  seen,  that  powers 
and  roots  may  be  multiplied  by  a  common  rule.  This  is  one 
of  the  many  advantages  derived  from  the  notation  by  frac- 
tional indices.  Any  quantities  whatever  may  be  reduced  to 
the  form  of  radicals,  (Art.  268.)  and  may  then  be  subjected 
to  the  same  modes  of  operation. 

Thus  y*xy'*=y3  +  *~=yl*.  [Art.  150.} 

I  JL.  i  •*"      «*  I 

And  x  X  x»  =x l  Ttl  =  xir. 

The  product  will  become  rational,  whenever  the  numera- 
tor of  the  index  can  be  exactly  divided  by  the  denominator. 


1  2  12 

T 

4 


Thus  «3  xa3  x  «7=  a  3  =a4.     (Art.  254.) 

And 

And 


130  ALGEBRA. 


282.  When  radical  quantities  which  arc  reduced  to  the 
same  index  have  rational  co-efficients,  the  rational  parts  may 
le  multiplied  together,  and  their  product  prefixed  to  the  product 
qf  the  radical  parts. 

1.  Multiply  ajb  into  c^/d. 

The  product  of  the  rational  parts  is  ac. 
The  product  of  the  radical  parts  is  -Jl)d.  (Art.  2T9.) 
And  the  whole  product  is  ac^/bd. 
For  a-y/6  is  «  x  -Jb.  (Art.  244.)     And  cjd  is  c  X  Jd.    » 
By  art.  102,  axV°  into-  ex  \/d,  is  ax  v^xex  jd;  or  by 
changing  the  order  of  .the  factors, 
Jd=ac  x 


2.  Multiply  ax3  into  Id'1. 

When  the  radical  parts  are  reduced  to  a  common  index, 

i  i 

the  factors  become  aT3 


The  product  then  is  ab(x*df')6'. 

But  in  cases  of  this  nature,  we  may  save  the  trouble  of 
reducing  to  a  common  index,  by  multiplying  as  in  art.  278. 

Thus  ax*  into  Id  *,  i 


•\r   i            fj          >i  —  i 

iViiut.   0(0+ x)*  aVy            QT/X            ax          x    T/3 

1  1^ 

Into      y(b—  x}'*  oVty          b^/x             by           y  3^9 

Prod.  ay(b 2  — a; 2 )5  ab^x z  —abx 


283.  If  the  rational  quantities,  instead  of  being  co-effi- 
cients to  the  radical  quantities,  are  connected  with  them  by 
the  signs  +   and    —  ,  each  term  in  the  multiplier  must  be 
into  each  in  the  multiplicand,  33  in  art.  100. 

Multiply  a 
Into 


cc+Cv/6  (Art.  244.) 

ajd+jbd  (Art.  279.) 


RADICAL  QUANTITIES. 

fThe  prodnct  of  «+  Vy  mt°  1  +rv/y  *s 


a       y  +  ariy    ?y    9r  ry 


DIVISION  OF  RADICAL  QUANTITIES. 

284.  The  division  of  radical  quantities  may  be  expressed, 
by  writing  the  divisor  under  the  dividend,  in  the  form  of  a 
fraction. 

V« 
Thus  the  quotient  of  3^/a  divided  by  \fl,  is  ~~n~" 

L      (a4-h}^ 
And   (*+*)     divided  by  (6+ff)"  is—  i-V 


In  these  instances,  the  radical  sign  or  index  is  separately 
applied  to  the  numerator  and  the  denominator.  But  if  the 
divisor  and  dividend  are  reduced  to  the  same  index  or  rad- 
ical sign,  this  may  be  applied  to  the  whole  quotient. 


Thus 


*V«      n   In 
«-i-  V^—  l^Th—    v  fT' 


tion  is  equal  to  the  root  of  the  numerator  divided  by  the  root 
of  the  denominator.  [Art.  260.] 

Again  n->/ab-~  n^b  =  n^a.  For  the  product  of  this  quo- 
tient into  the  divisor  is  equal  to  the  dividend,  that  is, 
V«  X  V&  =  nVao-  [Art.  279.]  Hence, 

285.  Quantities  under  the  saine  radical  sign  or  index,  may 
"be  divided  like  rational  quantities,  the  quotient  being  placed  un*- 
der  the  common  radical  sign  or  index. 

i          i 
Divide  (,r 3  y  2 )  '6~  by  y 2 . 

These  reduced  to  the  same  index  are  (xsy'2)w  and  (y*}6~* 
[Art.  269.] 

.1  -?  _1 

Aad  the  quotient  is  (x 3 ) 6  =  y?  =  x*.   [Art.  258.] 


Divide  v6a*#      vdhx*       (a 


'By        V3x         Vdx           a'J  (a*)'" 

Quot.     ^2a"3~  (f. 


132  ALGEBRA. 

28G.  A  root  is  divided  by  another  root  of  the  same  letter  er 
quantity,  by  subtracting  the  index  of  the  divisor  from  that  of 
tJie  dividend. 

«         i         i  _  i         3_i        s         i 
Thus  «*-,-  a*  =s  «a     *  -  a*     '*  —  a*  =  ,i   . 

•  1  3  1  1  1  J 

For  «'5  =  or  =  a75"  x  a¥  X  aff  and  this  divided  by  a*  -is 


0V 

.T.  JL          I__JL. 

Jn  the  same  manner,  it  may  be  shown  that  a™  -4-  a"  =  a"1 

Divide   (3rt)^      (a-r)7     /»m       (ft  +  y)«         (r*y3)^ 
By          £  (ax)?     a 


Quot.     (3a)^  «.'•  •"i'i»     (r2^3)     * 


'Powers  and  roofe  may  be  brought  promiscuously  togethcr; 
and  divided  according  to  the  same  rule.     See  art.  281. 

1  __  1  5  5.  TL  <L 

Thus  «*4-«7=aa     T=ft7.      For  a3  X  a3=a*=os. 

I-  _L 

So  yn  +  y'*=y*    '*. 

287.  When  radical  quantities  which  are  reduced  to  the 
same  index  have  rational  co-e.jjici.ents,  the  rational  parts  may 
be  divided  separately,  and  their  quotient  prefixed  to  the  quo-' 
tient  of  the  radical  parts. 

Thus  ac^/bd~-a^b=^c-^d.  For  this  quotient  multiplied 
into  the  divisor  is  equal  to  the  dividend.  [Art.  282.] 


Divide  ZAzJay      IQdti^b?      by(a*x*y      1CV32 
By         6  /a  2V*  yfa}*  8^/4 


b(a*xy 


1  1 

Divide  ab{x 2  b)  *  by  n(a')^, 


RADICAL  QUANTITIES.  133 

These  reduced  to  the  same  index  are  ab(x*b}~  and  a(x2)*' 

The  quotient  then  is  b(b)*  =  (b*  )*.  (Art.  272.) 

To  save  the  trouble  of  reducing  to  a  common  index,  the 

division  may  be  expressed  in  the  foftn  of  a  fraction,  as  in 

art.  284. 

«%*6)* 

The  quotient  will  then  be  — — ^ — • 


INVOLUTION  OF  RADICAL  QUANTITIES. 

288.  Radical  quantities,  like  powers,  are  involved  by 

the  index  of  the  root  into  the  index  of  the  required 
power. 

1  ^  v  2        2  ^         ^        J? 
1 .  The  square  of  a7  =  aT '      =  aT.     For  a"3  x  a?  —  a,3. 

[Art.  280.] 

2.  The  cube  of  a*  =  a*  X    =  a*.  For  a*  x  a*  X  a*  =  a7. 

3.  And  universally,  the  wth  power  of  a'"  =  a'" *n  —  flm'. 

i        1*1 
For  the  nth  power  of  am=  a'"  x  «TO n  times,  and  the 

n 

sum  of  the  indices  will  then  be  » . 

11  £     5 

4.  The  5th  power  of  o^jr5",  is  azy~s.      Or,  by  reducing 
file  roots  to  a  common  index, 

1  y  C  5 

.    i    i  .        -3    3.  3_ 

5.  The  cube  of  «"  of"  is  a"  x"1   or   (a™**  )•"». 

2  1        *  i. 
€.  The  square  of  a7 a:'1,  is  a3a?4. 

The  cube  of  a3  is  a^x3~  a?=a.  [Art.  253.] 

And  the  nth  power  of  an ,  is  an  =a.     That  is, 

289.  A  root  is  raised  lo  a  power  of  the  same  name,  by  re- 
moving the  index  or  radical  sign. 

Thus  the  cube  of   3v/J+lr,  is  b+x. 
A»d  the  nth  power  of  («—#)" ,  is  a—y. 


134  ALGEBRA. 

290.  When  the  radical  quantities  have  rational  coefficients, 
these  must  also  be  involved. 

I.  The  square  a  V^  's  fl2  "V^4- 

/cr=a3V*»-  [Art.  282.] 


2.  The  nth  power  of  «m,rm,isa'im,rm. 

3.  The  square  of  a  Vac— y,  is  a*  X  («—  y). 

4.  The  cube  of  3atyy,  is  27a3y. 

291.  But  if  the  radical  quantities  are  connected  with  oth- 
ers by  the  signs  -f  and  — ,  they  must  be  involved  by  a  mul- 
tiplication of  the  several  terms,  as  in  art.  213. 

Ex.  1.  Required  the  cube  of 


[Art.  244.] 
ajy+y    [Art.  289.] 


10 


2,  Required  the  square  of  a—^/b.     Ans.  «s— 

3.  Required  the  cube  of 


292.  It  is  unnecessary  to  give  a  separate  rule  for  the  evo- 
lution of  radical  quantities,  that  is,  for  finding  the  root,  of  a 
C[iu;nlity  which  is  already  a  root.  The  operation  is  the  same 
as  in  other  cases  of  evolution.  The  fractional  index  of 
the  radical  quantity  is  to  be  divided,  by  the  number  expres- 
sing the  root  to  be  found.  Or,  the  radical  sign  belonging  to 
the  required  root,  may  be  placed  over  the  given  quantity, 
f  Art.  257.]  If  there  are  rational  co-efficients,  the  roots  9f 
these  must  also  be  extracted. 


RADICAL  QUANTITIES.  135 

!     .          i.JL.9  1 

Thus,  the  square  root  of  a s,  is  a*  '     =  a8".    , 

1  i          i 

For  a*X  aT=a*. 
The   cube  root  of  «*,  is  a3 


The  nth  root  of  a* ^ by,  is    V  tt  * 

293.  It  may  be  proper  to  observe,  that  dividing  the  frac- 
tional index  of  a  root  is  the  same  in  effect,  as  multiplying  the 
number  which  is  placed  over  the  radical  sign.  For  this  num- 
ber corresponds  with  the  denominator  of  the  fractional  index; 
and  a  fraction  is  divided,  by  multiplying  its  denominator. 
[Art.  163.] 


On  the  other  hand,  multiplying  the  fractional  index  is 
equivalent  to  dividing  the  number  which  is  placed  over  the 
radical  sign. 

Thus  the  square  of  *  -/a  or  a*,  is  *  y«  or  a*  x    =  #!* 


^ 


SECTION 


REDUCTION  OF  EQUATIONS  BY  INVOLUTION 
AND  EVOLUTION. 

™ 

A        294     1^  an  e(lu^tlon>  tnc  letter  which  espressos  the 
unknown  quantity  is  sometimes  found  under 
a  radical  sign.     We  may  have 

To  clear  this  of  the  radical  sign,  let  each  member  of  the 
equation  be  squared,  that  is,  multiplied  into  itself.  We  shall 
then  have 

-v/^X  ^x—aa,        Or,  [Art.  289:]  x=az. 

The  equality  of  the  sides  is  not  affected  by  this  operation, 
because  each  is  only  multiplied  into  itself,  that  is,  equal  quan- 
ties  are  multiplied  into  equal  quantities.  [Ax.  3.] 

The  same  principle  is  applicable  to  any  root  whatever. 
If  '•  Jx—a ;  then  x—an .  For  by  art.  289,  a  root  is  raised  to 
a  power  cf  the  same  name,  by  removing  the  index  or  radi- 
cal sign.  Hence, 

295.  When  the  unknown  quantity  is  under  a  RADICAL  SIGN, 
the  equation  is  reduced  by  INVOLVING  both  sides,  to  a  power  of 
the  same  name,  as  the  root  expressed  by  the  radical  sign. 

It  will  generally  be  expedient  to  make  the  necessary  trans- 
positions, before  involving  the  quantities;  so  that  all  those 
which  are  not  under  the  radical  sign,  may  stand  on  one  side 
.     of  the  equation. 

Ex.   1.   Reduce  the  equation  ^4-4=9 

Transposing  -f4[Art,  173.]  <Jx— 9— 4=5 

Involving  both  bides  x  —5  2  =25. 

2.  Reduce  the  equation  «+  V*— b=d 

By  transposition,  ni/x=d+b—u 

By  involution,  ,v  =.-.  (d-{-  b—a)n' 

i 


feQOATIONS.  137 


1  Reduce  the  equation 

Involving  both  sides,  x+  1  =4S  =64 

Transposing  +1,  #=65. 

4.  Reduce  the  equation  A  +  2t/x—  4  =  6  -f  ? 
Clearing  of  fractions,  [Art.  183.]  8+6^*—  4=13 

Transposing  8,  6^—4=13-8=5 

Dividing  by  6,  [Art.  184.}  V#-4=£ 

Involving  both  sides,  *_4=f  f-  [Art.  223.] 

Transposing  —4,  x  =  f  f-  +  4. 


Reduce  the  equation' 


Multiplying  by  V 

Transposing  a2,  v^=3Tf  d—  a* 

Involving  both  sides,  #=(3-+df—  a2)3. 

In  the  first  step  in  this  example,  multiplying  the  first  mem- 
ber into  Va2  -f-  T/X,  that  is,  into  itself,  is  the  same  as  squaring 
rt,  which  is  cfone  by  taking  away  its  radical  sign.  The  oth- 
er member  being  a  fraction,  is  multiplied  into  a  quantity 
equal1  to  its  denominator,  by  cancelling  the  denominator. 
(Art.  159.)  There  remains  a  radical  sign  over  x,  which 
must  be  removed  by  involving  both  sides  01  the  equation. 

6.  Reduce  3+2^*—  4=6.        Ans.  o:=-f  &£. 


S/T  •* 


7.  Reduce 4-V/-F  =8'.  Ans.  #=20. 


REDUCTION  OF  EQUATIONS  BY  EVOLUTION. 

296.  In  many  equations,  the  letter  which  expresses  the 
unknown  quantity  is  involved  to  some  power.  Thus  in  th« 
equation 

#3=16 

we  have  the  value  of  the  square  of  x,  but  not  of  x  itself. 
.If  the  square  root  of  both  sides  be  extracted,  we  shalt  have 

#=4. 

The  equality  of  the  members  is  not  affected  by  tuls  r«- 
T 


138  ALGEBRA. 

duction.      For  if  two  quantities  or  sets  of  quantities  are 
equal,  their  roots  are  also  equal. 


If   (x+  a)"  =6+A,   then*+tf=V£+A.    Hence, 

297.  When  the  expression  containing  the  unknown  quantity 
is  a  POWER,  the  equation  is  reduced  by  EXTRACTING  the  ROOT 
of  both  sides,  a  root  of  the  same  name  as  the  power. 

Ex.  1.  Reduce  the  equation  S+*1*  —  8=7 

By  transposition  a?2  =7+8—  6=9 

By  evolution  *=±v/9=±3. 

The  signs  +  and  —  are  both  placed  before  V9,  be- 
cause an  even  root  of  an  affirmative  quantity  is  ambiguous. 
[Art.  261.] 

2.  Reduce  the  equation  £#*—  30=**  +34 
Trans,  and  uniting  terms,               4r2  =64 
Dividing  by  4,  -z*  =  16 

By  evolution,  #=±4. 

xa  x* 

3.  Reduce  the  equation       a+-v-=A—  -r 

Clearing  of  fractions,         abd+dx*  =bdJi—bx2 
Transposing  terms^  bx2  +  dx*  =  bdh  —atid 

bdh—abd 
Dividing  by  b  +  d  (Art.185.)   x*  =  -fe     ^ 


By  evolution  x  =  J_  ^  —  TT 


4.  Reduce  the  equation  «+<£*"  =10—  a* 

By  transposition,  dx"  +xn  =10—  a 

W-a 
Dividing  by  d+  1  ,  x*  = 


10-aU 
By  evolution, 


298.  From  the  preceding  articles,  it  will  be  easy  to  see  in 
what  manner  an  equation  is  to  be  reduced,  when  the  ex- 
pression containing  the  unknown  quantity  is  a  power,  and  at 
the  same  time  under  a  radical  sign;  that  is,  when  it  is  a  root 


EQUATIONS.  139 

of  a  power.    Both  involution  and  evolution  will  be  necessa- 
ry in  this  case, 

Ex.  1.  Reduce  the  equation  3v/#2:=4 

By  involution,  xz  =43  =64 

By  evolution,  a?=±  ^64  =±8. 

2.  Reduce  the  equation  Vxm—a=h—d 

By  involution  #"*— «=A2  —  2hd+d2 

Transposing  a  xm-h2  -2Jid+d2+a 

By  evolution  x=&h*  -2hd+d*+a. 


2      «+6 


3.  Reduce  the  equation         (#-f  a)  = £ 

Multiplying  by  (*-«)*  [Art.  279.]  (x*  -a*)?=a+l> 
By  involution,  x9  —a2  =a2  +2a&  +  62 

Trans,  and  uniting  terms,          x*  =2a2  +2aZ 


By  evolution  x  =±(2a2  +  2ab  +  6 


PROBLEMS. 

Prob.  1.  A  gentleman  being  asked  his  age,  replied;  "If 
you  add  to  it  ten  years,  and  extract  the  square  root  of  the 
sum,  and  from  this  root  subtract  2,  the  remainder  will  be  6." 
What  was  his  age  ? 


By  the  conditions  of  the  problem      "V^v+lO—  2=6 

By  transposition,  V#+10  =6+2=8 

By  involution,  #+10=  8*  =64 

By  transposition,  x  =64  —  1  0  =  54. 

Proof  [Art.  194.]     ^54+10-2=6. 

Prob.  2.  If  to  a  certain  number  22577  be  added,  and  the 
square  root  of  the  sum  be  extracted,  and  from  this  163  be 
subtracted,  the  remainder  will  be  237.  What  is  the  num- 
ber? 

Let  #=the  number  sought  £=163 

«=22577  « 


J40  ALGEBRA. 

By  the  conditions  proposed        V^+a—  b—c 

By  transposition,  ^x-\-a^=c-\-b 

By  involution,  x-\- a  =  (c-t-i)* 

By  transposition,  a;=(c+6)2  —  a 

Restoring  the  numbers,  (Art.52.)a:=(237+163)8  —22577 
That  is,  "  x  =  1 60000 — 22577  =  1 37423. 

JProof        V137423T22577- 163=237. 

299.  When  an  equation  is  reduced  by  extracting  .an  even 
joot  of  a  quantity,  the  solution  does  not  determine  whether 
the  answer  is  positive  or  negative.  [Art.  297.]  But  what  is 
ihus  left  ambiguous  by  the  algebraic  process,  is  frequently 
settled  by  the  statement  of  the  problem. 

,A  Prob.  3.  A  merchant  gains  in  trade  a  sum,  to  which  320 
dollars  bears  the  same  proportion,  as  five  times  this  sum  doe* 
to  2500.  What  is  the  amount  gained  ? 

Let  ,xi=the  sum  required. 
0=320 
£=2500. 

By  the  supposition  a :  #; :  5x :  b 

Mult,  extremes  and  means  [Art.  188.]     5x2  —ab 

ab 
Dividing  by  5,  x9  —-^ 

By  evolution, 

/ 320x2500  r 
Restoring  the  numbers,  x=  \ c —   ~)   =400. 

Here  the  answer  is  not  marked  as  ambiguous,  because  by 
4he  statement  of  the  problem  it  is  gain,  and  not  loss.  It 
must  therefore  be  positive.  This  might  be  determined,  in 
the  present  instance,  even  from  the  algebraic  process.  When- 
ever the  root  of  x*  is  ambiguous,  it  is  because  we  are  igno- 
rant whether  the  power  has  been  produced  by  the  multipli- 
cation of  +x,  or  of  —  x,  into  itsejf.  {Art.  262.]  But  here 
we  have  the  multiplication  actually  performed.  By  turning 
back  to  the  two  first  steps  of  the  equation,  we  find  that  5x* 
.was  produced  by  multiplying  5jf  into  x,  that  is  -f  5x  into 


x  =    r 


EQUATIONS.  i41 

Prob.  4.  The  distance  to  a  certain  place  is  such,  that  if 
56  be  subtracted  from  the  square  of  the  number  of  miles, 
Ihje  remainder  will  be  48.  What  is  the  distance  ? 

Let  x—tl\e  distance  required. 

By  the  supposition,  #2  —  96  =  48 

By  transposition,  x*  =48  +  96  =  144 

By  evolution,  x  =  ^  1 44 = 1 2. 

Prob.  5.  If  three  times  the  square  of  a  certain  number 
fre  divided  by  four,  and  if  the  quotient  be  diminished  by  12, 
the  remainder  will  be  180.  What  is  the  number:1 

3** 
By  the  supposition  -^-— 12=180 

Multiplying  by  4,  and  trans.        3x*  =720+48  =763 
Dividing  by  3,  #*  =256_ 

By  evolution,  x  —  V256 =16. 

Prob.  6.  What  number  is  that,  the  fourth  part  of  whose 
square  being  subtracted  from  8,  leaves  a  remainder  equal  to 
four?  Ans.  4. 

?$  '  • 

AFFECTED  QUADRATIC  EQUATIONS. 

300.  Equations  are  divided  into  classes,  which  are  distin- 
guished from  each  other,  by  the  power  of  the  letter  that  ex- 
presses the  unknown  quantity.  Those  which  contain  only 
the  first  power  of  the  unknown  quantity,  are  called  equations 
of  one  dimension,  or  equations  of  ihe  first  degree.  Those  in 
which  the  highest  power  of  the  unknown  quantity  is  a  square, 
are  called  quadratic,  or  equations  of  the  second  degree ;  those 
in  which  the  highest  power  is  a  cube,  equations  of  the  third 
degree,  fyc. 

Thus  x =0  +  6,  is  an  equation  of  the  first  degree. 

x*  —c,  and  x1  -\-ax=d,  are  quadratic  equations,  or 
equations  of  the  second  degree. 

x3=h,  and  x*+axz  -\-bx=d,  are  cubic  equations,  or 
equations  of  the  third  degree. 

301.  Equations  are  also  divided  into  pure  and  affected 


1E2:  ALGEBRA. 

equations.  A  pure  equation  contains  only  one  poiver  of  the 
•renown  quantity,  'ihis  may  be  the  first,  second,  third,  or 
any  other  power.  An  affected  equation  contains  different 
fowers  of  the  unknown  quantity.  Thus, 

=«?—&,  is  a  pure  quadratic  equation. 
x2+bx=d,  an  affected  quadratic  equation. 

x3  —b—  c,  a  pure  cubic  equation. 
x3+axz+bx=h,  an  affected  cubic  equation. 

A  pure  equation  is  also  called  a  simple  equation.  But  this 
ttema  has  been  applied  in  too  vague  a  manner.  By  some 
writers,  it  is  extenued  to  pure  equations  of  every  degree  :  by 
•Chers,  it  is  confined  to  those  of  the  first  degree. 

IB  a  pure  equation,  all  the  terms  which  contain  the  un- 
known quantity  may  be  united  in  one,  (Art.  185.)  and  the 
equation,  however  complicated  in  other  respects,  may  be 
leduced  by  the  rules  which  have  already  been  given.  But 
93  an  affected  equation,  as  the  unknown  quantity  is  raised  to 
aSvfperent  powers,  the  terms  containing  these  powers  can  not 
&e  united.  (Art.  230.)  There  are  particular  rules  for  the 
•eduction  of  quadratic,  cubic,  and  biquadratic  equations.  Of 
t&ese,  only  the  first  will  be  considered  at  present. 

302.  An  affected  quadratic,  equation  is  one  which  contains 
iflie  unknown  quantity  in  one  term,  and  the  square  of  that  quan- 
tity in  another  term. 

The  unknown  quantity  may  be  originally  in  several  terms 
rf  the  equation.  But  all  these  may  be  reduced  to  two,  one 
containing  the  unknown  quantity,  and  the  other  its  square. 

303.  It  has  already  been  shown  that  a  pure  quadratic  is 
lolved  by  extracting  the  root  of  both  sides  of  the  equation.  An 
tt/jfeeted  quadratic  may  be  solved  in  the  same  way,  if  the 
Member  which  contains  the  unknown  quantity  is  an  exact 
square*  Thus  the  equation 

x*+2ax+a*=b+h 

way  be  reduced  by  evolution.  For  the  first  member  is  the 
of  a  binomial  quantity.  [Art.  264.]  And  its  root  is 
Therefore, 


x+a=  v^6+A,  and  by  transposing  a, 


304.  But  it  is  not  often  the  case,  that  a  member  of  an  af- 
fected quadratic  equation  is  an  exact  square,  till  an  addition- 


QUADRATIC  EQUATIONS. 

al  term  is  applied,  for  the  purpose  of  making  the  requhsei 
reduction.     In  the  equation 


the  side  containing  the  unknown  quantity  is  not  a  compete 
square.  The  two  terms  of  which  it  is  composed  are  indeeat 
such,  as  might  belong  to  the  square  of  a  binomial  quant&y.. 
[Art.  214.]  But  one  term  is  wanting.  We  have  then  &» 
inquire,  in  what  way  this  may  be  supplied.  From  havifflg; 
two  terms  of  the  square  of  a  binomial  given,  Low  shall  we 
find  the  third? 

Of  the  three  terms,  two  are  complete  powers,  and  tTae 
other  is  twice  the  product  of  the  roots  of  these  power*  ; 
[Art.  214.]  or,  which  is  the  same  thing,  the  product  of  one 
of  the  roots  into  twice  the  other.  In  the  expression 


the  term  %ax  consists  of  the  factors  2a  and  x.  The  latter  is  tfbe: 
unknown  quantity.  The  other  factor  2a  may  be  considered! 
the  co-efficient  of  the  unknown  quantity  ;  a  co-efficient  bciag 
another  name  for  a  factor.  [Art.  41.]  As  x  is  the  root  of  lUws 
first  term  xz  ;  the  other  factor  2a  is  twice  the  root  of  ^tfae 
third  term,  which  is  wanted  to  complete  the  square.  Thene- 
fore  half  2a  is  the  root  of  the  deficient  term,  and  «z  is  iSbe 
term  itselC,  The  square  completed  is 


where  it  will  be  seen  that  the  last  term  a  z  is  the  square  «sf 
half  2a,  and  2a  is  the  co-efficient  of  x  the  root  of  the  fis* 
term. 

In  the  same  manner,  it  may  be  proved,  that  the  last  -team 
of  the  square  of  any  binomial  quantity,  is  equal  to  ±ke 
square  of  half  the  co-efficient  of  the  root  of  the  first  term, 
From  this  principle,  is  derived  the  following  rule  ? 

305.  To  complete  the  square,  in  an  affected  quadratic  eqioa- 
tion  ;  take  the  square  of  half  the  co-efficient  of  the  first  power 
of  the  unknown  quantity,  and  add  it  to  both  sides  of  the  equatim- 

Before  completing  the  square,  the  known  and  unknown* 
quantities  must  be  brought  on  opposite  sides  of  the  equals* 
by  transposition  ;  and  the  highest  power  of  the  nnkaowat 
quantity  must  have  the  affirmative  sign,  and  be  cleared  saf 
fractions,  co-efficients,  &,c.  See  arts.  308,  9,  10,  11, 

After  the  square  is  completed,  the  equation  is  reduced,  fey 
extracting  the  square  root  of  both  sides,  and  transposing  tibe 
known  part  of  the  binomial  root,  [Art.  303.] 


l4'i  ALGEBRA. 

The  quantity  wliich  is  added  to  one  side  of  the  equation', 
to  complete  the  square,  must  be  added  to  the  other  side  al- 
so, to  preserve  the  equality  of  the  two  members.  (Ax.  1.) 

306.  To  avoid  having  the  attention  divided  among  too 
>•'"'  many  objects,  the  learner  should  distinguish  between  what  is 
peculiar  in  the  reduction  of  quadratic  equations,  and  what  is 
common  to  this  and  the  other  kinds  which  have  already  been 
considered.  The  peculiar  part,  in  the  resolution  of  affected 
quadratics,  is  the  completing  of  the  square.  The  other  steps 
are  similar  to  those  by  which  pure  equations  are  reduced. 

For  the  purpose  of  rendering  the  completing  of  the  square 
familiar,  there  will  be  an  advantage  in  beginning  with  exam- 
ples in  which  the  equation  is  already  prepared  for  this  step. 

Ex.  1.  Reduce  the  equation  x*  +  Qax=b 

Completing  the  square  x*  -f6a#+9a2  =9a2  +b 

Extracting  both  sides    (Art.303.)  y+3a=±v/9ar+6 
Transposing  3a  a=  —  3a±V9as  +b 

Here  the  co-efficient  of  x,  in  the  first  step,  is  6a; 

The  square  of  half  this  is  9a2,  which  being  added  to  both 
sides  completes  the  square.  The  equation  is  then  reduced 
by  extracting  the  root  of  each  member,  in  the  same  manner 
as  in  art.  297,  excepting  that  the  square  here  being  that  of  a 
binomial,  its  root  is  found  by  the  rule'  in  art.  265. 


2.  Reduce  the  equation  x*  — 

Completing  the  square,          x  2  —  Qbx  +  1  6ft*  =166  3  -f  k 

Extracting  both  sides  x—  46=±  Vl662  +A 

Transposing  —46,  #=4&±  Vl6i*+A 

In  this  example,  half  the  co-efficient  of  x  is  45,  the  square 
of  which  IQb2  is  to  be  added  to  both  sides  of  the  equation. 


3.  Reduce  the  equation  x*-\*ax—l)+h 

a2     a2 
Completing  the  square^       a;8+aar-j-"7"=="T' 

By  evolution, 

a 

Transposing    - 


a 


QUADRATIC  EQUATIONS. 

Here  the  co-efficient  of  x  is  a,  half  of  which  is  -^,  whose 

a2 
square  is  -£-•  (Art.  223.) 

4.  Reduce  the  equation  #*  —  x—h—  d 

Completing  the  square,         K*  —  -a?+  \  =  ^  +  h  —  d 

Extracting  and  transp.  a?  =|±(  £  +  h  —  d  )"*. 

Here  the  co-efficient  of  a?  is  1,  its  half  is  £,  and  the  squar* 
of  this  is  i. 


5.  Reduce  the  equation  e     ,  _, 

Completing  the  square,         «8  -|-  3# + 1 = £  + d+ 6 

Extracting  and  transp.  *•=  — 1±(  %  +  d +6  )"*. 

6.  Reduce  the  equation         a?a  —  a&.v=a&  —  a? 
Completing  the  square,    #*  —  0&r-|-— ^— =— r—  -j-a&— cdf 

Extracting  and  transp.      <r =-^  "^  ( — -: — + ab —cdY* 

£  ~~  \     v  * 

ax 

7.  Reduce  the  equation         x*+-r=h 

ax     a  *       &  * 
Completing  the  square,      a?  *  +  -r- + TVF = 4}T  **"  ^ 

a  if  a*         >^, 
Extracting  and  transp,          a?=  —  £j~__  I JIT+^  /  , 

• 

By  art.  158,y  =yx^.    The  co-efficient  of  x,  therefore, 

a  a 

is  y    Half  of  this  is^,  (Art.  163.)  th£  square  of  which  is 


U 


ALGEBRA. 


8.  Reduce  the  equation,  x*  —~r  = 

Completing  the  square,          ^ 
Extracting  and  transp.  *=26—  \46»~"^7'v   • 


x      1 
Here  the  fraction  "T=^Xar.  (Art.  158.)     Therefore  the 

co-efficient  of  #  is  -r  • 
6 

307.  Tn  these  and  similar  instances,  the  root  of  the  third 
term  of  the  completed  square  is  easily  found,  because  this 
root  is  the  same  half  co-efficient  from  which  the  term  has 
just  been  derived.   (Art.304.)      Thus  in  the  last  example, 

half  the  co-efficient  of  x  is  51,  and  this  is  the  root  of  the 

m  ^ 

third  term  "JIT* 

308.  When  the  first  power  of  the  unknown  quantity  is  in 
several  terms,  these  should  be  united  in  one,  if  they  can  be, 
by  the  rules  for  reduction  in  additioa.     But  if  there  are  lite- 
ral co-efficients,  these  may  be  considered  as  constituting,  to- 
gether, a  compound  co-efficient  or  factor,  into  which  the  un- 
known quantity  is  multiplied. 

Thus  ax+bjc+dv=(a+b  +  d)xx.  (Art.  120.)  The 
square  of  half  this  compound  co-efficient  is  to  be  added  to 
both  sides  of  the  equation. 


1.  Reduce  the  equation 

Uniting  terms,  (Art  174.)  x*+6x—d 

Completing  the  square,  a?a+&r+9 

Extracting  and  transp.  #=  —  3± 


2.  Reduce  the  equation 

By  art.  120,  x*  +  (a+b)xx=h 

Compl'g  the  square,  x2  +(a+b}xx+(~-)   =(~^~) 


a+b  j 

By  evolution,  x^~~:i=^~ 


a+b,     //rt+6 
By  transposition,  x=-——^_  V  \-^~ 


QUADRATIC  EQUATIONS.  147 

3.  Reduce  the  equation,     a.2+or— ot—l 
By  art.  120,  s*  +  (a— 1)  X*=& 

(a-iy     A»-l\3 
€omprg the  square,  x* +(a-l}xx4-\— -g-f    =  \2    /    ^ 

fl-i+  ITa^ 

Extracting  and  transp.*  =  —  •  A,"" _  v  \~2 

309.  After  becoming  familiar  with  the  method  of  com- 
pleting the  square,  in  affected  quadratic  equations,  it  wiU  be 
necessary  to  attend  to  the  steps  which  are  preparatory  to-, 
tiik.  Here  however,  little  more  is  necessary,  than  an  appli- 
cation of  rules  already  given.  The  known  and  unknown 
quantities  must  be  brought  on  opposite  sides  of  the  equation 
by  transposition.  And  it  will  generally  be  expedient  to 
make  the  square  of  the  unknown  quantity  the  first  or  lead- 
ing term-,  as  in  the  .preceding  examples.  This  indeed  is  not 
essential.  But  it  will  show,  to  the  best  advantage,  the  ar- 
rangement of  the  terms  in  the  completed  square. 

1.  Reduce  the  equation  a+5x— 3b=3x— x* 

Transp.  and  uniting  terms,     x2  +  2#=36  — a 
Completing  the  square,         cr2+2Ar+l=l+3&  — « 

Extracting  and  transp.          x—  —  1±  V 1  +  3&  _  «. 
~»if<'  siH  "*v  far bob 

x       36 
2.  Reduce  the  equation  ~2~'  -i-2~^ 

Clearing  of  fractions,  xz+2x=12— 8x— 16 

Transp.  and  uniting  terms,    x*  +  I0a;=56 
Completing  the  square,        x2  4-10^+25=25  +  56=81 
Extracting  and  transp.         #=  —  5±  v'Sl  =  —  5±9. 


31 0.  If  the  highest  power  of  the  unknown  quantity  has  any 
co-efficient  or  divisor,  it  must,  before  the  square  is  completed, 
be  freed  from  these,  by  multiplication  or  division,  as  in  arts. 
180  and  184. 

I.  Reduce  the  equation  #2+24a— 6A  =  12x1— 

Transp.  a,nd  uniting  terms,    Go:2—  12-r=6A  —  24a 
Dividing  by  €,  x  2  —  2# = h — 4a 

Completing  the  square,         a,13— 2r+l=l+A— 4« 
Extracting  and  tran sp .          /r  =  It  V 1  +  h — 4a 


. 


146  ALGEBRA. 

bx* 
JL.  Reduce  the  equation     A-f  2#=rf—  -- 

Clearing  of  fractions,      ah  +  2ax=ad—  bx* 
By  transposition,  bx*  +%ax—ad—  oh 

2ax  _  ad—  ah 
Dividing  by  b, 


.Compl'g  the  square, 

«_!_/«* 
Extracting  and  transp.      x  =  —  T~  _  I  yj  -J- 

311.  If  the  square  of  the  unknown  quantity  is  in  several 
terms,  the  equation  must  be  divided  by  all  the  co-efficients 
of  this  square,  as  in  art.  185. 

1.  Reduce  the  equation         bx*+dz*—  4r=6—  h 

4*      b  -h 
Dividing  by  b+d,  (Art. 


2   \2      /   2    \2   b-h 
Completing  the 


2 
Extract,  and  transp          * 


2.  Reduce  the  equation 

Ti-ansp.  and  uniting  terms,     ax*  -\-x*  —2x=h 

2x          h 
Dividing  hy  a-f-1, 

Completing  the 

i     . 

Extracting  and  transp.  *=-_I 


312.  In  the  square  of  a  binomial,  the  first  and  last  terms 
Are  alvyays  positive.  For  each  is  the  square  of  one  of  the 
terms  of  the  root.  (Art.  214.)  But  every  square  is  positive. 
.(Art.  £15.  j  If  then  —  x3  occurs  in  an  equation,  it  can  not, 
with  this  sign,  form  a  part  of  the  square  far  a  binomial.  But 
£f  att  the  signs  in  the  equation  «be  changed,  the  equality  of 
the  sides  will  be  preserved,  (Art.  177.)  the  term  —x?  will 
•Ibecoioe  positive,  and  the  square  may  be  completed. 


QUADRATIC  EQUATIONS.  U9 

1.  Reduce  the  equation  —  xz  +2x=d—  h 
Changing  all  the  signs,  x*—2x=h—d 
Completing  the  square,  x2  —2x-{-l=l  -\-h-d 
Extracting  and  transp.  •   #=l±vi-j-A  —  d. 

~*~ 

2.  Reduce  the  equation  4r—  x3  =  —  12 
Changing  all  the  signs,  x*  —  4^  =  12 
Completing  the  square,  xz  —  4a:-j-4=4+  12=16 
Extractin    and  trans.  # 


xtracting  and  transp. 

313.  In  a  quadratic  equation,  the  first  term  x*  is  the 
square  of  a  single  letter.  But  a  binomial  quantity  may  con- 
sist of  terms,  one  or  both  of  which  are  already  powers. 

Thus  x*  +  a  is  a  binomial,  and  its  square  is 


where  the  index  of  x  in  the  first  term  is  twice  as  great  as  in 
the  second.  When  the  third  term  is  deficient,  the  square 
jnay  be  completed  in  the  same  manner  as  that  of  any  other 
binomial.  For  the  middle  term  is  twice  the  product  of  the 
roots  of  the  two  others. 

So  the  square  of  xn  -f  #,  is  x^+Zax*  +a2.      Therefore, 

314.  Any  equation  which  contains  only  two  different  powers 

of  the  unknown  quantity,  the  index  of  one  of  which  is  twice 

that  of  the  other,  may  be  resolved  in  the  same  manner  as  a 

quadratic  equation,  by  completing  the  square. 

It  must  be  observed  however  that,  in  tli€  binomial  root, 
the  letter  expressing  the  unknown  quantity  will  still  have  an 
index,  so  that  a  farther  extraction,  according  to  art.  297,  will 
Joe  necessary. 

Reduce  the  equation  x*  —  xz  =5  —  a 

Completing  the  square,         x4"—  x2+^=±  +  b  —  a 
Extracting  and  transp.  x*  =±±.V±-\-l>  —  a 

Extracting  again,  (Art.297.)  a?=  /|±  V 


>—  a. 


2.  Reduce  the  equation  x*n— Abx*  =a 

Completing  the  square,  x*n— &uxn  +4J2  =4i2  +a 

Extracting  and  transp.  x"  =21+  \/462  +a 

i->  "  -\l 

i^xtr&ctma;  a2;am  x=    V25 


,315.  The  solution  of  a  quadratic  equation,  whether  pure 


150  ALGEBRA. 

or  affected,  gives  two  results.  For  after  the  equation  is  re- 
duced, it  contains  an  ambiguous  root.  In  a  pure  quadratic, 
this  root  is  the  whole  value  of  the  unknown  quantity.  (Art. 
297.) 

Thus  the  equation  x*  =64 

Becomes,  when  reduced,  x =iv'64 

That  is,  the  value  of  x  'u  either  +8  or  —8,  for  each 
of  these  is  a  root  of  64.  Here  both  the  values  of  x  are  the 
same,  except  that  they  have  contrary  signs.  This  will  -be 
the  case  in  every  pure  quadratic  equation,  because  the  whole 
of  the  second  member  is  under  the  radical  sign.  The  two 
values  of  the  unknown  quantity  will  be  alike,  except  that  one 
will  be  positive,  and  the  other  negative. 

316.  But  in  affected  quadratics,  a  part  only  of  one  side  of 
the  reduced  equation  is  under  the  radical  i  ign.      When  this 
part  is  added  to,  or  subtracted  from,  that  which  is  without  the 
radical  sign  ;  the  two  results  will  differ  in  quantity,  ..and  will 
have  their  signs  in  some  cases  alike,  and  in  others  unlike. 

1.  The  equation  x3  +  8* =20 
Becomes,  when  reduced,  x=  —  4±  ^16+20 
That  is,                     .Ar.'.  #=-4±6. 

Here  the  first  value  of  x  is,  —  4+6  =  +  2  ~)    one  positive,  and 
And  the  second  is,  —4—6  =  — 10  jj  the  other  negative. 

2.  The  e  quation  a?  —  Sx  —  —  1 5 

Becomes,  when  reduced,         x—  4±  Vl6  — 15 
That  is  #=4±1 

Here  the  first  value  of  x  is  4+ 1  =  4-5),    ,u 

And  the  second  is  4-1  =  +  3  \  botb  Posltjve' 

That  these  two  values  of  x  are  correctly  found,  may  be 
proved,  by  substituting  first  one,  and  then  the  other,  hrx  it- 
self, in  the  original  equation.  (Art.  194.) 

Thus  55  -8X5  =25-40= -15 
And   32 -8x3=9-24= -15. 

317.  In  the  reduction  of  an  affected  quadratic  equation, 
the  value  of  the  unknown  quantity  is  frequently  found  to  be 
imaginary. 


QUADRATIC  EQUATIONS.  15! 

Thus  the  equation  *8  —  Sx=  —  20 

Becomes,  when  reduced,  #=4±  V16— 20 

That  is,  x^£  V~^A. 

vyyt- 

Here  the  root  of  the  negative  quantity  —4  can  not  be  as- 
signed, (Art.  263.)  and  therefore  the  value  of  x  can  not  be 
found.  There  will  be  the  same  impossibility,  in  every  in- 
stance in  which  the  negative  part  of  the  quantities  under  the 
radical  sign  is  greater  than  the  positive  part.* 

318.  Whenever  one  of  the  values  of  the  unknown  quanti- 
ty, in  a  quadratic  equation  is  imaginary,  the  other  is  so  also. 
For  botli  are  equally  affected  by  the  imaginary  root. 

Thus,  in  the  example  above, 

The  first  value  of  x  is  4+^—4, 

And  the  second  is  4— •/— 4;  each  of  which 
contains  the  imaginary  quantity  •/— 4. 

319.  An  equation  which  when  reduced  contains  an  ima- 
ginary root,  is  often  of  use,  to  enable  us  to  determine  wheth- 
er a  proposed  question  admits  of  an  answer,  or  involves  an 
absurdity. 

Suppose  it  is  required  to  divide  8  into  two  such  parts,  that 
the  product  will  be  20. 

If  x  is  one  of  the  parts,  the  other  will  be  8— ^.(Art.195.) 

By  the  conditions  proposed,  (8— #)  Xx=2Q 

That  is,  8x-x*  =20 

Changing  all  the  signs,  (Art.  177.)  x2  —  8x  =  —  20 

This  becomes,  when  reduced,  *'=4±v/— 4. 

Here  the  imaginary  expression  -J  —&  shows  that  an  an- 
swer is  impossible ;  and  that  there  is  an  absurdity  in  suppos- 
ing that  8  may  be  divided  into  two  such  parts,  that  their  pro- 
duct shall  be  20. 

320.  Although  a  quadratic  equation  has  two  solutions,  yet 
both  these  may  not  always  be  applicable  to  the  subject  pro- 
posed.    The  quantity  under  the  radical  sign  may  be  produ- 
ced either  from  a  positive  or  a  negative  root.      But  both 
these  roots  may  not,  in  every  insteice,  belong  to  the  prob- 
lem to  be  solved.     See  art.  299. 

*  See  note  G. 


152  ALGEBRA. 


PROBLEMS  PRODUCING  QUADRATIC  EQUATIONS. 

Prob.  1.  A  merchant  has  a  piece  of  cotton  cloth,  and  a 
piece  of  silk.  The  number  of  yards  in  both  is  110;  and  if 
the  square  of  the  number  of  yards  of  silk  be  subtracted 
from  80  limes  the  number  of  yards  of  cotton,  the  difference 
will  be  400.  How  many  yards  are  there  in  each  piece  ? 

Let  ,v=the  yards  of  silk. 
Then  110— *v=the  yards  of  cotton. 

By  supposition,  400=80  x(110-a;)-^ 

That  is,  400=8800— 80*— x* 

Transp.  &  unit.terms,  #*  +  80* = 8400 

CoinpPgthe  square,  x*  +  80* +1600  =  1 600 +8400  =  10000 

Extracting  and  transp.  x—  —  40tVToOOO  =  — 40±100 

The  first  value  of  .*,  is  —40+100=60,  the  yards  of  silk; 

And  110— #  =  110—  60=50,  the  yards  of  cotton. 

The  second  value  of  .v,  is  —40  — 100=  — 140;  but  as  this 
is  a  negative  quantity,  it  is  not  applicable  to  goods  which  a 
man  has  in  his  possession. 

Prob.  2.  The  ages  of  two  brothers  are  such,  that  their 
sum  is  45  years,  and  their  product  500.  What  is  the  age  of 
each  ? 

Let  *=one  of  the  ages.     Then  45— #=the  other. 

By  supposition,  x  x  (45 — x]  =500 

That  is,  Aox—x*  =r500 

Changing  all  the  signs,  cc2  —45.*=  —500 

2025     2025     -          25 
Compl'g  the  square,     x*  —  45#+— 7- •=— ~r~— 500=-^ 

45  .     /25    45      5 
Extract,  and  transp.      #=^r-\/~r=~7r-";1  ' 

2  —  V    4         _        Z 

One  of  the  ages  then  is  25  years,  and  the  other  20. 

Prob.  3.  To  find  two  numbers  snch,  that  their  difference 
shall  be  4,  and  their  product  117. 

Let  #=one  number,  and  #+4=the  other. 


QUADRATIC  EQUATIONS. 

By  the  conditions,  (x + 4)  x  x  =  11 7 

This  reduced,  gives,  *•  =  —  2±  V  12T=  — 2±1  1. 

One  of  the  numbers  therefore  is  9,  and  the  other  13. 

Prob.  4.  A  merchant  having  sold  a  piece  of  cloth  which 
cost  him  30  dollars,  found  that  if  the  price  for  which  he  sold 
it  were  multiplied  by  his  gain,  {he  product  would  be  equal 
to  the  cube  of  his  gain.  What  was  his  gain  ? 

Let  #=the  gain. 
Then  30+^=the  price  for  which  the  cloth  was  sold. 

By  the  statement,  x3  =(30+*-)  x  x 

That  is,  x3=3Qx+x* 

Dividing  by  a?,  (Art.  186.)  x2  =30+# 

Transposing*1,  x*  —  #=30 

Completing  the  square,  x2  — #  + £=^  +  30 

Extracting  and  transposing  x = ^—  V i  .4-  30 = i— 

The  first  value  of  x  is  3-+IsL  =+6 
The  second  value  is     £  —  -,*-  =  — 5 

As  the  last  answer  is  negative,  it  is  to  be  rejected  as  incon- 
sistent with  the  nature  of  the  problem,  (Art.  320.)  for  gain, 
must  be  considered  positive. 

Prob.  5.  To  find  two  numbers,  whose  difference  shall  be 
3,  and  the  difference  of  their  cubes  117. 

Let  tfssthe  least  number. 
Then  x  -f  3  =  the  greatest! 

By  supposition,  (x + 3) 3  -  x 3  =  ll  7 

Expanding  (x + 3) 3  (Art.21 7.)  9*  *  +27x  =  1 1 7  -  27 = 90 
Dividing  by  9,  #*  +  3*  =  1 0 

Completing  the  square,  x*  +  3* + 1  ==£  + 1 0  =  4* 

Extracting  and  transp.  x  =  — 1±  V  Y  =  — 11| . 

The  two  numbers,  therefoie,.  are  2  and  5. 

Prob.  6.  To  find  two  numbers,  whose  difference  shall  b« 
12,  and  the  sum  of  their  squares  1424. 

Aas.  Th«  numbers  are  20  and  33. 
V 


154  ALGEBRA. 

Protr.  7.  Two  persons  draw  prizes  in  a  lottery,  the  differ- 
ence of  which  is  120  dollars,  and  the  greater  is  to  the  Jess, 
4s  the  less  to  10.  What  are  tke  prizes  f 

Let  :sr=the  less  prize. 
Then  v  4- 120  =  the  greater. 

By  the  statement,  x -f  1 20 :  x : :  \  :  1 0 

Mult,  extremes  and  means,  x*  =10*+ 1200 

Transposing  1 0#,  *»  —  1 0*  =  1 200 

Completing  the  square,  *8  —  1  Ox +25=25 -f  1200 

Extracting  and  transp.  x=5+  V25 4-1200=5  + 35, 

The  two  prizes,  then,  are  40  and  160. 

Prob*  8.  What  two  numbers  are  those  whose  sum  is  6, 
and  the  sum  of  their  cubes  72?  /"^ 

Are.  2  and  4. 


.... 


. 


SUBSTITUTION 


321.  In  the  redaction  of  Quadratic  EqtrationSj  as  welt  as 
m  other  parts  of  algebra,  a  complicated  process  may  be  ren- 
dered much  more  simple,  by  introducing  a  new  tetter  which 
shall  be  made  to  represent  several  others.  This  is  termed 
substitution.  A  letter  may  be  put  for  a  compound  quantity 
as  well  as  few  a  single  number.  Thus  in  the  equation 

x*  -2a#=|4VS6 -64+ A, 

we.  may  substitute  b,  for  |  +  -v/86— 64-f  A.      The  equation 
will  then  become  **  —  2ax=b,  and  when  reduced 

will  be  x—aL  ^a*  +b. 

After  the  operation  is  completed,  the  compound  quantity 
for  which  a  single  letter  has  been  substituted,  may  be  restor- 
ed. The  last  equation,  by  restoring  the  value  of  b,  will 
become 


=<r±  A8  +  A  +  ^86-64+ A, 


QUADRATIC  EQUATIONS.  15$ 

Reduce  the  equation  ax—  2x—  d=bx—  xs  —  x 

Transp.  and  uniting  terms,  x2  +ax—  bx—  x—d 
By  art.  120,  x2  +(a-6-l)  xx=d 

SubstitutingAfor(a—  6  —  I),*2  +hx=d 

Completing  the  square, 


Extracting  and  transp.          «•==—  "2_ 

H-6-1+    Ra^pW 
Restoring  the  value  of  A,    AT  =  —  --  ^       —  v 


SECTION  XI. 


SOLUTION    OF    PROBLEMS    WHICH    CONTAIN 

TWO  OR  MORE  UNKNOWN  QUANTITIES. 

DEMONSTRATION  OF  THEOREMS. 


AR  322  T^  ^e  e*amples  which  have  been  given  of  the 
resolution  of  equations,  in  the  preceding  sec- 
tions, each  problem  has  contained  only  one.  unknown  quan- 
tity. Or  if,  in  some  instances,  there  have  been  two,  they 
have  been  so  related  to  each  other,  that  both  have  been  ex- 
pressed by  means  of  the  same  letter.  (Art.  195.) 

But  cases  frequently  occur  in  which  several  unknown 
qaantities  are  introduced  into  the  same  calculation.  And  if 
the  problem  is  of  such  a  nature,  as  to  admit  of  a  determi- 
nate answer,  there  will  arise  from  the  conditions,  as  many 
equations  independent  of  evach  other,  as  there  are  unknown 
quantities. 

Equations  are  said  to  be  independent,  when  they  express 
different  conditions  ;  and  dependent,  when  they  express  the 
same  conditions  under  different  forms.  The  former  are  not 
convertible  into  each  other.  But  the  latter  may  be  chan- 
ged from  one  form  to  the  other,  by  the  methods  of  reduc- 
tion which  have  been  considered.  Thus  b—x=y,  and/>  =y-\-x, 
are  dependent  equations,  because  one  is  formed  from  the 
other  by  merely  transposing  x. 

323.  In  solving  a  problem,  it  is  necessary  first  to  find  the 
value  of  one  of  the  unknown  quantities,  and  then  of  the 
others  in  succession.  To  do  this,  we  must  derive  from  the 
.equations  which  are  given,  a  new  equation,  from  which  all 
the  unknown  quantities  except  one  shall  be  excluded. 

Suppose  the  following  equations  are  given. 


%.  x—  y=2. 

If  y  be  transposed  in  each,  they  will  become 
1.  #  =  14— 


EQUATIONS.  157 

Here  the  first  member  of  each  of  the  equations  is  x,  and 
the  second  member  of  each  is  equal  to  x.  But  according  to 
axiom  5th,  quantities  which  are  respectively  equal  to  any 
other  quantity  are  equal  to  each  other;  therefore, 


Here  we  have  a  new  equation,  which  contains  only  the 
unknown  quantity  y. 

Transposing  2  and  — y,  2y=12 

Dividmg  by  2,  y=6. 

The  value  of  y  is  therefore  found.     Hence, 

324.  RULE  J.  To  exterminate  one  of  two  unknown  quan- 
tities, and  deduce  one  equation  from  two ;  Find  the  value  of 
one  of  the  unknown  quantities  in  each  of  the  equations,  and 
form  a  new  equation  by  making  one  of  these  values  equal  to  the 
ether. 

That  quantity  which  is  the  least  involved  should  be  the 
one  which  is  chosen  to  be  exterminated. 

For  the  convenience  of  referring  to  different  parts  of  a 
solution,  the  several  steps  will,  in  future,  be  numbered. 
When  an  equation  is  formed  from  one  immediately  preceding, 
it  will  be  unnecessary  to  specify  it.  In  other  cases,  the  num- 
ber of  the  equation  or  equations  from  which  a  new  one  is 
derived  will  be  referred  to. 

Prob.  1 .  To  find  two  numbers  such,  that 
Their  sum  shall  be  24 ;  and 
The  greater  shall  be  equal  to  5  times  the  less. 

Let  #— the  greater;  And  y=the  less. 

1 .  By  the  first  condition,  .xi+y=24) 

2.  By  the  second,  x=5y        ) 

3.  Transp.  y  in  the  1st  equation,  #=24— y 

4.  Making  the  2d  and  3d  equal,  5y=24— y 

5.  Transp.  and  uniting  terms,  6y=24 

6.  Dividing  by  6,  y=4,  the  less  number. 

JProb.  2.  To  find  one  of  two  quantities, 
Whose  sum  is  equal  to  h ;  and 
The  difference  of  whose  squares  is  equal  to  d. 

Ar=the  greater  quantity;  And  y— the  less. 


JSS  ALGEBRA. 

1.  By  the  first  condition,  x+y=h     > 

2L  By  the  second,  AT*  — y*  =d  £ 

3i  Transp.  ya  in  the  2d  equation,  x* 

4,  By  evolution,  (Art.  297.)  x— 

5.  Transp.  y  in  the  1st  equation,    ac=h—y 

C.  Making  the  4th  and  5th  equal     Vd+ y*=h— 

7.  By  involution,  (Art.  295.)  d+y*  =A»  —  2 

8.  Expunging  jr  (Art.  176.)  d—hz—2hy 
9;  By  transposition,  2hy=hz—d 

h*  —  d 
10.  Dividing  by  2A,  y~~~ ' 


Pfcob.  3.  Given  ax+by—h  >  Ji—ad 

And        x+y=d  V     To  find  y-  Ans.  y  =-r— r ' 


325.  The  rule  giren  above  may  be  generally  applied,  for 
the  extermination  of  unknown  quantities.  But  there  are 
cases,  in  which  other  methods  will  be  found  more  expedi- 
fkuis. 

Suppose  x=hy 
And     ax+  bx=y* 

As  in  the  first  of  these  equations  x  is  equal  to  Ay,  we  may, 
in  the  second  equation,  substitute  this  value  of  x  instead  of  x 
ifcelC.  The  second  equation  will  then  be  converted  into 


Tlie  equality  of  the  two  sides  is  not  affected  by  this  alter- 
ationy  because  we  only  exchange  one  quantity  x,  for  another 
which/  is  equal  to  it.  By  this  means  we  obtain  an  equation 
which  contains  only  one  unknown  quantity.  Hence, 

326,  RUI*E.  II.  To  exterminate  an  unknown  quantity, 
Find  the  value  of  one  of  the  unknown  quantities,  in  one  of  the 
equations  ;  and  then,  in  the  other  equation,  SUBSTITUTE  this 
value,  for  the  unknown  quantity  itself. 

Prob.  4.  A  privateer  in  chase  of  a  ship  20  miles  distant, 
sails  8  miles,  while  the  ship  sails  7.  How  far  must  the  pri- 
vateer sail,  before  she  overtakes  the  ship  ? 

It  is  erident  that  the  whole  distance  which  the  privateer 
sails  during  the  chase,  must  be  to  the  distance  which  the  ship 
sails  in  the  same  time,  as  8  to  7. 


EQUATIONS.  159 

Let  #=the  distance  which  the  privateer  sails; 
And  y  =the  distance  which  the  ship  sails* 

1.  By  the  supposition,  ;r=y-{-20  ) 

2.  And  also,  ,r:y::8:7> 

3.  Mult,  extremes  and  means,  8y=7x 

4.  Dividing  by  8,  y—-$x 

5.  Substituting  \*fory  in  the  Isteq.  x=%x 

6.  Multiplying  by  8,  and  transp.          x=lQO. 

L^ 

Prob.  5.  The  ages  of  two  persons  A  and  B  are  such,  that 
seven  years  ago,  A  was  three  times  as  old  as  B;  and  seven 
years  hence,  A  will  be  twice  as  old  as  B.  What  is  the  age 
of  B? 

Let  *=the  age  of  A;          And  y=the  age  of  B  ; 
Then  x—1  was  the  age  of  A,  7  years  ago; 
And    y—  7  was  the  age  of  B,  7  years  ago. 
Also  x+1  will  be  the  age  of  A,  7  years  hence; 
And  y+1  will  be  the  age  of  B,  7  years  hence. 

1.  By  the  first  condition,  x—  7=3x  (y—  7)=3y—  21  f 

2.  By  the  second,  *+7=2x(y+7)=2y+14  $ 

3.  Transp.  7  in  the  1st  equa.      <r=3y—  14 

4.  Subst.3y-14forcr,inthe2d,  3y—  14+7=%+14 

5.  Transp.  and  uniting  terms,  y=2l,  the  age  of  B. 

Prob.  6.  There  are  two  numbers,  of  which 

The  greater  is  to  the  less,  as  3  to  2  ;  and 
Their  sum  is  the  sixth  part  of  their  product. 

What  is  the  less  number  ?  Ana.  10. 

327.  There  is  a  third  method  of  exterminating  an  un- 
known quantity  from  an  equation,  which,  in  many  cases,  is 
preferable  to  either  of  the  preceding. 

Suppose  that  x-\-3y=a  > 
And  that        x—  3^=5) 

If  we  add  together  the  first  members  of  these  two  equa- 
tions, and  also  the  second  members,  we  shall  hare 


an  equation  which  contains  only  the  unknown  quantity  *. 
The  other,  having  equal  co-efficients  with  contrary  signs, 
has  disappeared.  (Art.  77.)  The  equality  of  the  sides  is 


ICO  ALGEBRA, 

preserved,  because  we  have  only  added  equal  quantities  lv 
equal  quantities.  (Ax.  1.) 

Again  suppose  3r-fy=A  ) 
And  2x+y=d\ 

If  we  subtract  the  last  equation  from  the  first,  we  shall 
have 

x=h-d 

where  y  is  exterminated,  without  affecting  the  equality  of  the 
sides.  (Ax.  2.) 

Again,  suppose    x—  ly—a  ) 
And  £+4y=6  \ 


Multiplying  the  1st  by  2,        2x—  Ay=2a 

Then  adding  the  2d  and  3d,  3x=b  +  2a.      Hencer 


RULE  III.     To  exterminate  an  unknown  quantity, 
^MULTIPLY  or  DIVIDE   the  equation,  if  necessary,  in  such  a 

manner  that  the  term  ivhich  contains  one  of  the  unknown  quan- 

ties  shall  be  the  same  in  both. 

Then  SUBTRACT  one  equation  from  the  other,  if  the  signs  of 

this  unknown  quantity  are  ALIKE,  or  ADD  them  together,  if  the 

signs  are  UNLIKE. 

It  must  be  kept  in  mind  that  both  members  of  an  equa- 

tion are  always  to  be  increased  or  diminished,  multiplied  or 

divided  alike.  (Art.  170.) 

Prob.  7.  The  numbers  in  two  opposing  armies  are  sudhr 
that, 

The  sum  of  both  is  21110;  and 

Twice  the  number  in  the  greater  army,  added  to  three 
times  the  number  in  the  less,  is  52219. 

What  is  the  number  in  the  greater  army  ? 

Let  *-=the  greater.  And  y=the  less. 

1.  By  the  first  condition,  «4-y=21110      ) 

2.  By  the  second,  2v+3?/=52219  > 

t.  Multiplying  the  1st  by  3,  3#+3y=63330 

,  Subtracting  the  2d  from  the  3d,  x  =  1  1  1  1  1  . 

Prob.  8.  Given  2«-fy=16,  and  3*—  3y=6,  to  find  the 
value  of  #. 


EQUATIONS. 


1.  By  supposition,  2^+y  =  16). 

2.  And  3*-3#=6  5 

3.  Multiplying  the  1st  by  3,  6tf+3y=48 

4.  Adding'the  2d  and  3d,  9#=54 

B.  Dividing  by  9,  x=Q. 

a    j     > 

Prob.  9.  Given  x+  y—  14,  and  #  —  y=2,  to  find  the  value 
•f  y.  Ans.  6. 

In  the  succeeding  problems,  either  of  the  three  rules  for 
exterminating  unknown  quantities  will  be  made  use  of,  as 
will  in  each  case  be  most  convenient. 

329.  When  one  of  the  unknown  quantities  is  determined, 
the  other  may  be  easily  obtained,  by  going  back  to  an  equa- 
tion which  contains  both,  and  substituting,  instead  of  that 
which  is  already  found,  its  numerical  value. 

Prob.  10.  The  mast  of  a  ship  consists  of  two  parts  : 

One  third  of  the  lower  part,  added  to  one  sixth  of  the  up- 
per part,  is  equal  to  28  ;  and 

Five  times  the  lower  part,  diminished  by  six  times  the  up- 
per part,  is  equal  to  12. 

What  is  the  height  of  the  mast  ? 

• 

Let  ;c=the  lower  part  ;  And  y  =the  upper  part. 

1.  By  the  first  condition,  ^x-\-^y=2Q 

2.  By  the  second,  5x—tiy  =  l2 

3.  Multiplying  the  1  st  by  6,  2*  +  1/  =  1  68 

4.  Dividing  the  2d  by  6,  ^x—y=2 

5.  Adding  the  3d  and  4th,  2x  +  1  -x  =  1  70 

6.  Multiplying  by  6,  12r+5* 

7.  Uniting  terms,  and  dividing  by  17,  ^=60,the  lower  part. 

Then  by  the  3d  step,  2#+y=168 

That  is,  substituting  60  for  x,     120+y  =  168       [per  part. 

Transposing  120,  ^  =  168-1.20=48,  the  up- 


Prob.  11.  To  find  a  fraction  such 

If  a  unit  be  added  to  the  numerator,  the  fraction  will  be 
equal  to  -|  ;  but 

If  a  unit  be  added  to  the  denominator,  the.  fraction  will 
be  equal  to  -^-. 

Let  A-=the  numerator,  And  y=the  denominator. 

W 


162.  ALGEBRA. 

#+1 

1.  By  the  first  condition, 

y 

.V 

2.  By  the  second, 

3.  Clearing  the  1st  of  fractions,  3 

4.  Subst.  3* +3,  for  y  in  the  2d,  STT^* 

5.  Clearing  of  fractions,  4.x1 = 3* -j-^ 

tt.  Transp.  and  uniting  terms,     *=4,  the  numerator 

Then  subst.  4 for  x  in  the  3d,  1 2+3 = 1 5  =y,the  denominator. 

Prob.  12.  What  two  numbers  are  those, 

Whose  difference  h  to  their  sum,  a&  2  to  3  j  and 
Whose  sum  is  to  their  product,  as  3  to  5  ? 

Ans.  10  and  2, 

Prob.  13.  To  find  two  numbers  such,  that 

The  product  of  their  sum  and  difference  shall  be  5,  and 
The  product  of  the  sum  of  their  squares  and  the  differ- 
ence of  their  squares  shall  be  65., 

Let  ,r=the  greater  number  f  And  y=the  less. 

1.  By  the  first  condition,  (x+y)  X  (x— y)  =  5 

2.  By  the  second,  fc'+y8)  X  (*a  — ya)=65 

3.  Mult,  the  factors  in  the  1st,  (Art.  235.)     x2  — y1  =5 

4.  Dividing  the  2d  by  the  3d,  (Art,  118.)     za+y3  =13 

5.  Adding  the  3d  and  4th,  2*2=l8 

6.  Dividing  by  2,  x*  =9 

7.  By  evolution,  #=^9=3,  the  greater  number. 
The  other  number  is         2. 

In  the  4th  step,  the  first  member  of  the  3d  equation  is  di- 
vided by  x*  — y*,  and  the  second  member  by  5,  which  is 
equal  to  x*—y*. 

Prob.  14.  To  find  two  numbers,  whose  difference  is  8,  and 
product  240. 

Prob.  15.  To  find  two  numbers, 

Whose  difference  shall  be  12,  and 
The  sum  of  their  squares  1424. 

Let  #=the  greater;  And  y= the  lew. 


EQUATIONS.  163 

1.  By  the  1st  condition,  x—y*=12 

2.  By  the  second,  x*  +yz  — 1424 

3.  Transp.  y  in  the  1st,  x  =#+12 

4.  Squaring  both  sides,  xz  =y*  +2% +144 

5.  Transp.  t/8  in  the  2d,  a;3  =1424- J/3 
.6.  Making  the  4th  and  5th  equal,  ?/3+24?/+144  = 
7.  Transp.  and  uniting  terms,  2yz+24y  =  12 
•8.  Dividing  by  2,  «/ 2  + 1 2y = 640 
9.  Completing  the  square,  ^»  +  12y+_3Gj= 

1 Q.  Extracting  and  transp.  y  —  —  6±  VG76  =  —  G±26. 

And    ,v=y+ 12=20+ 12=32 


EQUATIONS  WHICH  CONTAIN   THREE    OR    MORE   UNKNOWN 
QUANTITIES. 

330.  In  the  examples  hitherto  given,  each  fens  contained 
no  more  than  two  unknown  quantities.  And  two  indepen- 
dent equations  have  been  sufficient  to  express  the  conditions 
of  the  question.  But  problems  may  involve  three  or  more 
unknown  quantities;  and  may  require  for  their  solution  as 
many  independent  equations. 

Suppose  x+y+z=12     ") 

And        x + 2y — 2z  =  10  >  are  given,  to  find  x.  y,  and  ,r. 

And         x+y— z=4        ) 

From  these  three  equations,  two  others  may  be  derived, 

/which  rliall  contain   only  two  unknown  quantities.      One  of 

the  three  in  the  original  equations  may  be  exterminated,  in 

the  same  manner  as  when  there  are,  at  first,  only  two,  by  the 

rules  in  arts.  324,  6,  8. 

In  the  equations  given  above,  if  we  transpose  y  and  ;r,  we 
shall  have, 

fn  the  first,  £  =  12— y— z  } 
In  the  second,  #=10— 2y-\- 2z  > 
In  the  third,  z—A— y+z  ) 

From  these  -we  may  deduce  two  new  equations,  froia 
which  x  shall  be  excluded. 

By  making  the  1st  and  2d  equal,    ]2—y—z—W—2y-\-2z  \ 
By  making  the  2d  and  3d  equal,     10— 2y-\-2.z  ==4— y+z    ) 

fly  trans,  and  uniting  terms,  in  the  1st  of  these  two,  y=3^-2  > 
By  trans,  and  uniting  tennis,  in  the  second,  y=.z-i-G  ) 


JG4  ALGEBRA. 

From  these  two  equations,  one  may  be  derived  contaia* 
ing  only  one  unknown  quantity, 

Making  one  equal  to  the  other,  3s  —  2=~-f6 

Transp.  uniting  terms,  and  dividing,  z=4.     Hence, 

331.  To  solve  a  problem  containing  three  unknown  quan- 
tities, and  producing  three  independent  equations, 

First,  from  the  three  equations  deduce  two,  containing  only 
two  unknown  quantities, 

Then,  from  these  two  deduce  one,  containing  only  one,  un- 
known quantity. 

For  making  these  reductions,  the  rules  already  given  are 
sufficient.  (Art.  324,  6,  8.) 

Prob.  16.  Let  there  be  given, 

1.  The  equation  x+5y+  6z=  53  ) 

2.  And  #+3y  +  3r=30  \  To  find  x,  y,  and  z. 

3.  And 


From  these  three  equations  to  derive  two,  containing  only 
two  unknown  quantities, 

4.  Subtract  the  2d  from  the  1st, 

5.  Subtract  the  3d  from  the  2d, 
From  these  two,  to  derive  one, 

6.  Subtract  the  5th  from  the  4th,    z=5. 

To  find  x  and  y,  we  have  only  to  take  their  values  from 
the  3d  and  5th  equations.  (Art.  329.) 

7.  Transp.  the  5th  and  dividing,   y=Q—  2=9  —  5=4 

8.  Transposing  in  the  3d,  x  =  1  2-z-y  —  1  2-5-4  =  3. 

Prob.  17.  To  find  x,  y,  and  z,  from 

1.  The  equation  x+y+z  —  12     "i 

2.  And  x+2y+3z=20'}- 

3.  And  £a?+!y+*=6    ) 

4.  Multiplying  the  1  st  by  3,  3*  +  3y  +  3z  =  36 

5.  Subtracting  the  2d  from  the  4lh,    2r4-?/  =  16 

6.  Subtracting  the  3d  from  the  1st,     x—  \x-\-y—  yy=6 

7.  Clearing  the  6th  of  fractions,         4^+  3?/  =$&&  ) 

8.  Multiplying  the  5th  by  3,  G.v+3y=48  $ 

9.  Subtracting  the  7th  from  the  8th,  2r  =  12.  And  #=6. 

36-4*     36-24 

10.  Transp.  in  the  7th,  and  dividing,  ?/=—-  ^—  =—-—  —  =4, 

jll.  Transp.  in  the  1st  equation,       y=12-,v-y=  12-6-4  =2. 


EQUATIONS. 

In  this  example  all  the  reductions  have  been  made  accor- 
ding to  'the  third  rule  for  exterminating  unknown  quantities. 
(Art.  328.)  But  either  of  the  three  may  be  used  at  pleas- 
ure. 

332.  A  calculation  may  often  be  very  much  abridged,  by 
the  exercise  of  judgment,  in  stating  the  question,  in  select- 
ing the  equations  from  which  others  are  to  be  deduced,  in 
simplifying  fractional  expressions,  in  avoiding  radical  quanti- 
ties, &c.  The  skill  which  is  necessary  for  this  purpose,  how- 
ever, is  to  be  acquired,  not  from  a  system  of  rules  ;  but 
from  practice,  and  a  habit  of  attention  to  the  peculiarities  in 
the  conditions  of  different  problems,  the  variety  of  ways  in 
which  the  same  quantity  may  be  expressed,  the  numerous 
forms  which  equations  may  assume,  &c.  In  many  of  tlje 
examples  in  this  and  the  preceding  sections,  the  processes 
might  have  been  shortened.  But  the  object  has  been  to  il- 
lustrate general  principles,  rather  than  to  furnish  specimens 
of  expeditious  solutions.  The  learner  will  do  well,  as  he 
passes  along,  to  exercise  his  skill  in  abridging  the  calcula- 
tions which  are  here  given,  or  substituting  others  in  their 
stead. 


fl.  *+y=«) 

Prob.  18.  Given,  <J  2.  x  +  z=b  j  To  find  x,  y,  and  z. 

(3.  y+z=c  5 

a+ly^-c  «+c—  b  b  +  c—a 

Ans.   x  =  ---  jc  ---  Andy=—  —  ^  —        And  z=  —  aT~* 

Prob.  19.  Three  persons  A.  B,  and  C,  purchase  a  horse 
for  100  dollars,  but  neither  is  able  to  pay  for  the  whole.  The 
payment  would  require, 

The  Avhole  of  A's  money,  together  with  half  of  B's  ;  or 

The  whole  of  B's,  with  one  third  of  C's  ;  or 

The  whole  of  C's,  with  one  fourth  of  A's. 
How  much  money  had  each? 

Let  x=  A's  z—C's 

=B's  «=100 


I6G  ALGEBRA. 

1 .  By  the  first  condition, 

2.  By  the  second, 

3.  By  the  third, 

4.  Transp.  in  the  1st,  and  clear,  fractions,  y =2a— 2r 

5.  Transp.  in  the  second,  */=«— }z 

6.  Making  the  4th  and  5th  equal,  2a— C2x  =  a  —  \z 

7.  Trans,  in  the  6th,  and  clear,  fractions,  z=Qx  —  3a  > 

8.  Trans,  in-  the  3d,  z=a— \x    ) 

9.  Making  the  7th  and  8th  equal,  6x— 3a=a— \* 

10.  Trans,  in  the  9th  and  clear,  fractions,  25z  =  16^  =  1600 

11.  Dividing  by  25,  x= 64,  A's  money. 
12  By  the  Sth  equation,         x—a—^x^lQO— 16=84,  C's. 
13.  By  the  Sth,                     y=a -£z=100-28=72,  BV 

333.  The  learner  must  exercise  his  own  judgment,  as  to 
the  choice  of  the  quantity  to  be  first  exterminated.  It  will 
generally  be  best  to  begin  with  that  which  is  most  free  from 
co-efficients,  fractions,  radical  signs,  Sic. 

Prob.  20.  The  sum  of  the  distances  which  three  persons, 
A,  B,  and  C,  have  travelled  is  62  miles ; 
A's  distance  is  equal  to  4  times  C's,  added  to  twice  B's;  and 
Twice  A's  added  to  3  times  B's,  is  equal  to  17  times  C's. 

What  are  the  respective  distances  ? 

Ans.  A's,  46  miles  ;  B's,  9 ;  and  C's  7. 

Prob.  21 .  To  find  -r,  y,  and  #,  from 

1.  The  equation  |a?+Ty+£s=62) 

2.  And  ^+^ 

3.  And  ^+ji/ 

1  4.  Clear,  the  1st  of  fractions,    12r+Sy+6z  = 

5.  Do.     the2d,  20a?+15y+ 12s =2820  \ 

6.  Do.     the  3d,  30x+24y +  20^=4560  ) 

7.  Mult,  the  4th  by  2,  24*+  16y  +  12z=2976 

8.  Subtract.  5th  from  7th,          4^+y=156 

9.  Mult,  the  5th  by  5,  100*-j-75y+60*=1410Q 

1 0.  Mult,  the  6th  by  3,  90* + 72y + 60s = 1 3680 

11.  Subtract.  10th  from  Sth,       10x+3y=420 

12.  Transp.  .in  the  Sth  ^=156—  4;c 

420— 1  Or 

13.  Do.  llth,  and  divid.  by  3,  y= s 

420-10* 

14.  Mak.  I2thand  13th  equal, s =156—4* 

15.  Clearing  of  fractions,  Sic.    £=24 

16.  Bythel2th,  y  =  156 -4*  =  156 -96 =60. 

17.  By  the  4th,  transp.  fcc.        z=120. 


EQUATIONS.  10t 

Cxy=QOQ) 

Prob.  22.  Given   <  a?z=300  >  To  find  x,  y,  and  z. 
(  yz 

S.  *=30. 


334.  The  same  method  which  is  employed  for  the  reduc- 
tion of  three  equations,  may  be  extended  to  4,  5,  or  any 
number  of  equations,  containing  as  many  unknown  quanti- 
ties. The  unknown  quantities  may  be  exterminated,  one  af- 
ter another,  and  the  number  of  equations  may  be  reduced 
by  successive  steps,  from  five  to  four,  from  four  to  three, 
from  three  to  two,  &c. 

Prob.  23.  To  find  w}  z,  y,  and  z,  from 

1.  The  equation  ^-y+z+^w 

2.  And  x+y+w=9 

3.  And  *+y+*=12 

4.  And  a?-j-w-fz=lO 

5.  Clear,  the  1st  of  frac.  y+2z+w=lQ 

6.  Subtract.  2d  from  3d,          z  —  w=3    >  TArce  equations; 

7.  Subtract.  4th  from  3d,         y—w=2    ) 

8.  Addins  5th  and  6th,       y+  32=19),™ 

9.  Subtract.  7th  from  6th,  -y+z  =  l  \  Two 

10.  Adding  8ih  and  9th,  4z=20.     Or  z=5      ^ 

11.  Transp.  in  the  8th,     y  =  19-3^  =  19-15  =4  1  Quantities 

12.  Transp.  in  the  3d,      cc  =  12—  y—  z—S          [required. 

13.  Transp.  in  the  2d,      w—Q—  x—  y=2          j 

*Prob.  24.  Given    <j  ^+  {20=2^     To  ^^  M>'  *•  ^  and  * 
[z+195=3w 

Answer.  t(?=100 
» 


. 


335.  If  in  the  algebraic  statement  of  the  conditions  of  a 


aec 

son' 

son's  Algebra,  Book  11,  Sec.  1,  Saundersori's  Algebra,  Book  li  and  m, 

and  Dodson's  Mathematical  Repository. 


163  ALGEBRA. 

problem,  the  original  equations  are  more  numerous  than  the 
unknown  quantities  ;  these  equations  will  either  be  contra- 
dictory, or  one  or  more  of  them  will  be  superfluous. 

Thus  the  equations  <  i-^Ion  (  are  contratuctory. 

For  by  the  first  ;r=»:20,  while  by  the  second  ar=40. 

But  if  the  latter  be  altered,  so  as  to  give  to  x  the  same  val- 
ue as  the  former,it  will  be  useless,  in  the  statement  of  a  prob- 
lem. For  nothing  can  be  determined  from  the  one,  \\hicli 
can  not  be  from  the  other. 

Thus,  of  the  equations  <  ,*~  1Q  >  one  is  superfluous. 

For  either  of  them  is  sufficient  to  determine  the  value  of 
x.  They  are  not  independent  equations.  (Art.  322.)  One  is 
convertible  into  the  other.  For  if  we  divide  the  1st  by  6,  it 
will  become  the  same  as  the  second. 

Or  if  we  multiply  the  second  by  6,  it  will  become  the 
same  as  the  first. 

336.  But  if  the  number  of  independent  equations  produ- 
ced from  the  conditions  of  a  problem,  is  less  than  the  num- 
ber of  unknown  quantities,  the  subject  is  not  sufficiently  lim- 
ited to  admit-of  a  definite  answer.  For  each  equation  can 
limit  but  one  quantity.  And  to  enable  us  to  find  this  quan- 
tity, all  the  others  connected  with  it,  must  either  be  previ- 
ously known,  or  be  determined  from  other  equations.  If 
this  is  not  the  case,  there  will  be  a  variety  of  answers  which 
will  equally  satisfy  the  conditions  of  the  question.  If,  for 
instance,  iu  the  equation 


x  and  y  are  required,  there  may  be  fifty  different  answers. 
The  values  of  x  and  y  may  be  either  99  and  1,  or  98  and  2, 
or  97  and  3,  &#.  For  the  sum  of  each  of  these  pairs  of 
numbers  is  equal  to  100.  But  if  there  is  a  second  equation 
which  determines  one  of  these  quantities,  the  other  may 
then  be  found  from  the  equation  already  given.  As  oc+y  =  100, 
if  x  =46,  y  must  be  such  a  number  as  added  to4C  will  make 
100,  that  is,  it  must  be  54.  No  other  number  will  answer 
this  condition. 

337.  For  the  sake  of  abridging  the  solution  of  a  problem,how- 
ever,  the  number  of  independent  equations  actually  put  upon 
paper  is  frequently  less,  than  the  number  of  unknown  quan- 


EQUATIONS.  169 

titles.  Suppose  we  are  required  to  divide  100  into  two  such 
parts  that  the  greater  shall  be  equal  to  three  times  the  less. 
If  we  put  x  for  fhe  greater,  the  less  will  be  100—  x.  (Art.196.) 

Then  by  the  supposition,  #=300  —  3x 

Transposing  and  dividing,  #=75,  the  greater. 

And  1 00  -  75 = 25,  the  less. 

Here,  two  unknown  quantities  are  found,  although  there 
appears  to  be  but  one  independent  equation.  The  reason  of 
this  is,  that  a  part  of  the  solution  has  been  omitted,  because 
it  is  so  simple,  as  to  be  easily  supplied  by  the  mind.  To 
have  a  view  of  the  whole,  without  abridging,  let  #=the 
greater  number,  and  y=the  less. 

1 .  Then  by  supposition,  x  -f  y = 1 00  > 

2.  And  3y=x          > 

3.  Transp.  x  in  the  1st,  y  —  lQQ—x 

4.  Dividing  the  2d  by  3,  y=^x 

5.  Making  the  3d  and  4th  equal,      |#  =  100  — x 

6.  Multiplying  by  3,  #=300  —  ox 

7.  Transp.  and  dividing,  #=75,  the  greater. 

8.  By  the  3d  step,  y=100  — #=25,  the  less. 

By  comparing  these  two  solutions  with  edNi  other,  it  will 
be  seen  that  the  first  begins  at  the  6th  step  of  the  latter,  all 
the  preceding  parts  being  omitted,  because  they  are  too  sim- 
ple to  require  the  formality  of  writing  down. 

Probi  To  find  two  numbers  whose  sum  is  30,  and  the 
difference  of  their  squares  120. 

Let  a =30  5  =  120 

#=the  less  number  required. 
•  Then  a— #=the  greater.  (Art.  195,) 
And  a2  — 2a#+#2  =the  square  of  the  greater.  (Art.  214.) 
From  thi*  subtract  x2  the  square  of  the  less,  and  we  shall 
have  a2  —  2a#=the  difference  of  their  squares. 

1.  By  supposition,  b=a2—2ax 

2.  By  transposition,  2.ax—a*—  b 

az—b 

3.  Dividing  by  2a  x  =~~^ — 

302-120 

4.  Restoring  the  numbers,       x= — ^ — ^r-=13,  the  less, 

*  X  oU 

And  a— #=30  — 13  =  17,  the  greater. 


ALGEBRA. 

338.  In  most  cases  also,  the  solution  of  a  problem  which- 
contains  many  unknown  quantities  may  be  abridged,  by  par- 
ticular artifices  in  substituting  a  single  letter  for  several. 
(Art.  321  :) 

*  Suppose  four  numbers,  «,  x,-y,  and  z,  arc  required, 

The  sum  of  the  three  first  is  13 

The  sum  of  the  two  first  and  last  17 

The  sum  of  the  first  and  two  last  18 

The  sura  of  the  three  last  21 

Then    1.  «+*+}/  =  13' 
2. 
3. 

4. 


Let  S  be  substituted  for  the  sum  of  the  four  numbers,  diaf 
is,  for  u+.v+y+z.  It  will  be  seen  that,  of  Ujese  four  equa- 
tions, 

The  first  cdntains  all  the  letters  except  z,  that  is,  S—  z  =  l  3- 
The  second  contains  all  except  y,  that  is,  S—  y=17 

The  third  contains  all  except  .v,  that  is,  S—  #  =  18 

The  fourth  contains  all  except'  »,  that  is,  £—  «=21  ,- 

Adding  all  thflfe  eqaatioria  together,  we  have 
4S'—  t—  y—  x—  w=69 


Or 

But    S=(z+y+x+y)  by  substitution. 
Therefore,  4S-S=69,  that  is,  3S=-69,  and  5=231 

Then  putting  23  for  5,  in  the  four  equations  in  which  it  is?' 
first  introduced,  we  have 


j       fo         y=23-17=6- 

l  *}  x=23-18=5 

23—  M=r2l  [^=23-21=2: 

Contrivances  of  this  sort  for  facilitating  th«  solution"  of 
particular  problems,  must  be  left  to  be  furnished  for  the  oc.- 
easion,  by  the  ingenuity  of  the  learner.  They  are  of  a  na- 
ture not  to  be  taught  by  a  system  of  rules. 

339.  In  the  resolution  oY  equations  containing  several  un- 
^  known  quantities,  there  will  often  be  an  advantage  in  adopt- 
ing the  Ibllowing  method  of  notation. 

*  Ludlam's  Algebra,  art.  161.  c- 


171 

The  co-efficients  of  one  of  the  unknown  quantities  are 
represented, 

In  the  first  equation,  by  a  single  letter,  as  a. 
In  the  second,  by  the  same  letter  marked  with  an  accent,  as  a'. 
In  the  third,  by  the  same  letter  with  a  double  accent,  as  a", 

&c. 

The  co-efficients  of  the  other  unknown  quantities,  are 
represented  by  other  letters  marked  in  a  similar  manner;  as 
are  also  the  terms  which  consist  of  known  quantities  only. 

Two  equations  containing  the  two  unknown  quantities  x 
and  y  may  be  written  .thus, 

<tx+by=c 


Three  equations  containing  x,  y,  and  z,  thus, 

ax+by+cz=d 
ti'x+b'y+c'z=d' 
a"x-{-b"y  +  c"z—d" 

Four  equations  containing  x,y,  z,  and  u,  thus, 

.tue  +  by  +cz  -\~du-e 
a 


a'''x  +  b'"y  +  c'"z  +  d'"u—e'" 

The  same  letter  is  made  the  ea-efficient  of  the  same  un~ 
'•known  quantity,  in  different  equations,  that  the  co-efficients 
of  the  several  unknown  quantities  may  be  distinguished,  in 
any  part  of  the  calculation.  But  the  letter  is  marked  with 
different  accents,  because  it  actually  stands  for  different  quan- 
tities. 

Thus  we  may  put  a  =4,     a'=6,     a"=10,     a"'=203&c. 


y=c      > 

>y=c  I 

bb'v—cb' 


To  find  ihe  value  jof  x  and  y. 

1.  In  the  equation,  .c 

2L  And  ax+b'% 

3.  Multiplying  the  1st  by  6',(Art.328.)  ab'x+bb'i/- 

4.  Multiplying  the  2d  by  b,  ba'x-{-bb'y=bc 

5.  Subtracting  the  4th  from  the  3d,     ab'x— ba'x=cb'— be' 

«  cb'—bc   "I 

€.  Dividing  by  ab'— ba  (Art.  121.)     x—~i^T~7  \ 

ac'—ca    » 
By  a  similar  process,  y  -^'^-Ta7  J 


172  ALGEBRA. 

The  symmetry  of  these  expressions  is  well  calculated  to 
fix  them  in  the  memory.  The  denominators  are  the  same 
in  both ;  and  the  numerators  are  like  the  denominators,  ex- 
cept a  change  of  one  of  the  letters  in  each  term.  But  the 
particular  advantage  of  this  method  is,  that  the  expressions 
here  obtained  may  be  considered  as  general  solutions,  which 
give  the  values  of  the  unknown  quantities,  in  other  equations 
of  a  similar  nature. 

Thus  if  10^+6^=100  > 
And       40*+% =200  $ 

Then  putting  a  — 10  b—6  q 

a' =40  b' =4  fc'=200 

cb'-bc       100x4-6x200 
We  have  *'=^I_17=   10X4-6X40"==4' 

oc'— ca'      10x200—100x40 
y  ^^fo7 =~10  X  4-6~x40~ 

The  equations  to  be  resolved  may,  originally,  consist  of 
more  than  three  terms.  But  if  they  are  of  the  first  degree, 
and  have  only  two  unknown  quantities,  each  may  be  redu- 
ced to  three  terms  by  substitution. 


Thus  the  equation  dv— &x+hy— 617=7 

Is  the  same,  by  art.  120,  as       (d— &}x+(h— 6)y=m+8 

And  putting  a=d-^-4,  b—h— 6  c=m  +  8 

It  becomes  ax+by=c.* 

DEMONSTRATION  OF  THEOREMS. 

340.  Equations  have  been  applied,  in  this  and  the  prece- 
ding sections,  to  the  solution  of  problems.  They  may  be  em- 
ployed with  equal  advantage,  in  the  demonstration  of  theorems. 
The*  principal  difference,  in  the  two  cases,  is  in  the  order  in 
which  the  steps  are  arranged.  The  operations  themselves  are 
substantially  the  same.  It  is  essential  to  a  demonstration,  that 
complete  certainty  be  earned  through  every  part  of  the  pro- 

*For  the  application  ef  this  plan  of  notation  to  the  solution  of 
equations  which  contain  more  than  two  unknown  quantities,  see  La 
Croix's  Algebra,  art.  85,  Maclaurin's  Algebra,  Part  i.  Chap.  12, 
Fenn's  Algebra,  p.  57,  and  a  paper  of  Laplace,  in  the  Memoirs  of  the 
Academy  of  Sciences  for  1172. 


THEOREMS'  173 

cess.  (Art.ll.)  This  is  effected,  in  the  reduction  of  equations, 
by  adhering  to  the  general  rule,  to  make  no  alteration  which 
shall  affect  the  value  of  one  of  the  member^ without  equally 
increasing  or  diminishing  the  other.  In  applying  this  princi- 
ple, we  are  guided  by  the  axioms  laid  down  in  art.  63.  These 
axioms  are  as  applicable  to  the  demonstration  of  theorems, 
as  to  the  solution  of  problems. 

But  the  order  of  the  steps  will  generally  be  different.  In 
solving  a  problem,  the  object  is  to  find  the  value  of  the  un- 
known quantity,  by  disengaging  it  from  all  other  quantities. 
But  in  conducting  a  demonstration,  k  is  necessary  to  bring 
the  equation  to  that  particular  form  which  will  express,  in 
algebraic  terms,  the  proposition  to  be  proved. 

Ex.  1.  Theorem.  Four  times  the  product  of  any  two 
numbers,  is  equal  to  the  square  of  their  sum,  diminished  by 
the  square  of  their  difference. 

Let  #=the  greater  number,  «=their  sum, 

y=the  less,  d=their  difference. 

Demonstration. 

1.  By  the  notation  x+y—s  > 

2.  And  x—  y=d\ 

3.  Adding  the  two,  (Ax.  1.)  Zx=s-{-d 

4.  Subtracting  the  2d  from  the  1st,  2w=s—d 

5.  Mult.  3d  and  4th,  (Ax.  3.)  Axy  =  (s  +  d)x(s-d) 

6.  That  is,  (Art.  235.)  4*z/=s2  -dz. 

The  last  equation  expressed  in  words  is  the  proposition 
which  was  to  be  demonstrated.  It  will  be  easily  seen  that 
it  is  equally  applicable  to  any  two  numbers  whatever.  For 
the  particular  values  of  x  and  y  will  make  no  difference  in 
the  nature  of  the  proof. 


=  (8+6)2-(8-6)3=192. 
And   4xlOxG=r(10+6)2-(10-6)2=240. 
And   4xl2xlO  =  (12+10)?-(12—  10)2=480. 


Theorem  2.  The  sum  of  the  squares  of  any  two  numbers*, 
is  equal  to  the  square  of  their  difference,  added  to  twice 
their  product. 

Let  a:==the  greater,  rf=their  difference. 

y=the  less,  p—  their  product. 


174  ALGEBRA. 

Demonstration. 

!.  By  the  notion  x—  y= 

2.  And  xy=p 

a:1—  ** 


3.  Squaring  the  first, 

4.  Multiplying  the  2d  by  2,  2xy=2p 

6.  Adding  the  3d  and  4th,  AT*  +y*  =rf*  -f  2j>. 

TliuslOz+82=(lO-8)2+2xlOx«=164. 

341.  General  propositions  are  also  discovered,  in  an  expe- 
ditious manner,  by  means  of  equations.  The  relations  of 
quantities  may  be  presented  to  our  view,  in  a  great  variety 
of  ways,  by  the  several  changes  through  which  a  given  equa- 
tion may  be  made  to  pass.  Each  step  in  the  process  will 
contain  a  distinct  proposition. 

Let  s  and  d  be  the  sum  aud  difference  of  two  quantities 
x  and  y,  as  before. 


1.  Then  s 

2.  And  d=x— 

3.  Dividing  the  1st  by  2,  £*=»*+-& 

4.  Dividing  the  2d  by  2,  \d-\x  —  \y 

5.  Adding  the  3d  and  4th,  \s-\-  \d=\x-\-\x-=.x 

6.  Sub.  the  4th  from  the  3d,  -|s  -  \  d—  |y  +  $y  =y. 


That  is, 

Half  the  difference  of  two  quantities*,  added  to  half  their 
*MT»,  is  equal  to  the  greater  ;  and 

Half  their  difference  subtracted  from  half  th«ir  sum,  w  equal 
to  the  less. 


. 


RATIO  AND  PROPORTION.* 


A  142  T^ffi  design  of  mathematical  investigations,  19 
to  arrive  at  the  knowledge  of  particular 
quantities,  by  comparing  them  with  other  quantities,  either 
equal  to,  or^  greater,  or  less  than  those  which  are  the  objects 
cf  inquiry.  The  end  is  most  commonly  attained  by  means 
of  a  series  of  equations  and  proportions.  When  we  make 
use  of  equations,  we  determine  the  quantity  sought,  by  dis- 
eovering  its  equality  with  some  other 'quantity  or  quantities 
already  known. 

We  have  frequent  Occasion,  however,  to  compare  the  un- 
known quantity  with  others  which  are  not  equal  to  it,  but  ei- 
ther greater  or  less.  Here,  a  different  mode  of  proceeding  be- 
comes necessary.  We  may  inquire,  either  how  much  one  of 
the  quantities  is  greater  than  the  other  j  or  how  many  times 
the  one  contains  the  other.  la  finding  the  answer  to  either 
of  these  inquiries,  we  discover  what  is  termed  a  ratio  of  the 
two  quantities.  One  is  called  arithmetical,  and  the  other  gr~ 
^metrical  ratio.  It  should  be  observed,  however,  that  both 
these  terms  have  been  adopted  arbitrarily,  merely  for  dis- 
tinction sake.  Arithmetical  ratio,  and  geometrical  ratio, 
are  both  of  them  applicable  to  arithmetic,  and  both  to  ge- 
ometry. 

As  the  whole  of  tire  extensive  and  important  subjeet  of 
proportion  depends  upon  ratios,  it  is  necessary  that  theser 
should  be  clearly  and  fully  understood. 

343.  ARITHMETICAL  RATIO  is  the  DIFFERENCE  between  tw» 
quantities  or  sets  of  quantities.  The  quantities  themselves 
are  called  the  terms  of  the  ratio,  that  is,  the  terms  between 

*  Euclid's  Elements,  Book  5, 7.  3.  Euler's  Algebra,  Part  i.  Sec.  3. 
Emerson  on  Proportion.  Camus'  Geometry,  Book  in.  Ludlam's 
Mathematics.  Vfallis'  Algebra,  Chap.  19,  «0.  Saunderson's  Alge^ 
bra,  Book  7.  Barrow's  Mathematical  Lectures.  See  also  an  ingen- 
ious essay  on  the  5th  book  of  Eudid,  ia  lUti  Annlyst  for  March 
fey  Professor  Adram. 


17C  ALGEBRA. 

which  the  ratio  exists.  Thus  2  is  the  arithmetical  ratio  of 
5  to  3.  This  is  sometimes  expressed,  hy  placing  two  points 
between  the  quantities  thus  5  .  .3,  which  is  the  same  as  5—3. 
Indeed  the  term  arithmetical  ratio,  and  its  notation  by  points 
are  almost  needless.  For  the  one  is  only  a  substitute  for  the 
word  difference,  and  the  other  for  the  sign  — . 

344.  If  both  the  terms  of  an  arithmetical  ratio  be  multi- 
plied or  divided  by  the  same  quantity,  the  ratio  will,  in  effect, 
be  multiplied  or  divided  by  that  quantity. 

Thus  if  a-l-r 

Then  muU.  both  sides  by  A,  (Ax.  3.)     ha-~hb=hr 

a      b      r 
And  dividing  by  h,  (Ax.  4.)  -r  —  T~T " 

345.  If  the  terms  of  one  arithmetical  rati'o  be  added  to, 
or  subtracted' from,  the  corresponding  terms  of  another,  the 
ratio  of  their  sum  or  difference  will  be  equal  to  the  sum  of 
difference  of  the  two  ratios. 


Andrf-A]arethetworati°3' 


Then  (a+<7)-(6  +  h}  =  (a-b)  +  (d-h}.  For  each =a+d-l-!t. 
And    (a— d)  —  (b— h)  =  (a  — b)  —  (d-k).  For  each =a-d-b+h. 

Thus  the  arith.  ratio  of  11  ..4  is  7 
And    the  arith.  ratio  of    5 . .  2  is  3 

The  ratio  of  the  SUPI  of  the  termslG . .  Gisl  0,the  sum  of  the  ratios. 
The  ratio  of  the  diff.  of  the  terms  6 .  .2  is  4,the  diff.  of  the  ratio?. 

346.  GEOMETRICAL  RATIO  is  that  relation  between  quanti- 
ties which  ts  expressed  by  the  QUOTIENT  of  the  one  divided  by 
the  other. 

Thus  the  ratio  of  8  to  4,  is  |  or  2.  For  thig  is  the  quotient 
of  8  divided  by  4.  In  other  words,  it  shows  how  often  4  is 
contained  in  8. 

In  the  same  manner,  the  ratio  of  any  quantity  to  another 
may  be  expressed  by  dividing  the  former  by  the  latter,  or, 
which  is  the  same  thing,  making  the  former  the  numerator 
of  a  fraction,  and  the  latter  the  denominator. 

a 
Thus  the  ratio  of  a  to  b  is  -r- 

.  d+h 
The  ratio  of  d+  k  to  b+c,  isrr^T.  . 


RATIO.  177 

347.  Geometrical  ratio  is  also  expressed  by  placing  two 
points,  one  over  the  other,  between  the  quantities  com- 
pared. 

Thus  a  :  b  expresses  the  ratio  of  a  to  I ;  and  12 : 4  the 
ratio  of  12  to  4.  The  two  quantities  together  are  called  a 
couplet,  of  which  the  first  term  is  the  antecedent,  and  the  last, 
the  consequent. 

348.  This  notation  by  points,  and  the  other  in  the  form  of 
a  fraction,  may  be  exchanged  the  one  for  the  other,  as  con- 
venience may  require  ;  observing  to  make  the  antecedent  of 
the  couplet,  the  numerator  of  the  fraction,  and  the  conse- 
quent the  denominator. 

6 
Thus  10:5  is  the  same  as  y  and  b  :  d,  the  same  as -7  • 

349.  Of  these  three,  the  antecedent,  the  consequent,  and 
the  ratio,  any  two  being  given,  the  other  may  be  found. 

Let  «=the  antecedent,  c= the  consequent,  r=the  ratio. 

By  definition  r±=— ;  that  is,  the  ratio  is  equal  to  the  antece- 

c 

dent  divided  by  the  consequent. 

Multiplying  by  c,  «=cr,  that  is,  the  antecedent  is  equal  to 
the  consequent  multiplied  into  the  ratio. 

a 
Dividing  by  r,  c=— ,  that  is,  the  consequent  is  equal  to  the 

antecedent  divided  by  the  ratio. 

Cor.  1.  If  two  couplets  have  their  antecedents  equal,  and 
their  consequents  equal,  their  ratios  must  be  equal.* 

Cor.  2.  If,  in  two  couplets,  the  ratios  are  equal,  and  the 
antecedents  equalr  the  consequents  are  equal ;  and  if  the 
ratios  are  equal  and  the  consequents  equal,  the  antecedents 
are  equal.f 

350.  If  the  two  quantities  compared  are  equal,  the  ratio  is 
a  unit,  or  a  ratio  of  equality.     Thus  the  ratio  of  3x6  : 18  is 
a  unit,  for  the  quotient  of  any  quantity  divided  by  itself  is  1. 

If  the  antecedent  of  a  couplet  is  greater  than  the  con- 
sequent, the  ratio  is  greater  than  a  unit.  For  ii'  a  dividend 
is  greater  than  its  divisor,  the  quotient  is  greater  than  a  unit. 

*  Euclid  7.  5.  fEuc.  9.  .5.. 

Y 


1,78  ALGEBRA. 

Thus  the  ratio  of  18 :  G  is  3.  (Art.  128.  cor.)     This  is  called 
a  ratio  of  greater  inequality. 

On  the  other  hand,  if  the  antecedent  is  less  than  the  con- 
sequent, the  ratio  is  less  than  a  unit,  and  is  called  a  ratio  of 
less  inequality.  Thus  the  ratio  of  2 :  3,  is  less  than  a  unit,  be- 
cause the  dividend  is  less  than  the  divisor. 

351.  INVERSE  or  RECIPROCAL  ratio  is  the  ratio  of  the  re- 
ciprocals of  two  quantities.     See  art.  49. 

Thus  the  reciprocal  ratio  of  6  to  3,  is  |  to  •£,  that  is  £-7- -3. 

a 

The  direct  ratio  of  a  to  b  is  -r,  that  is,  the  antecedent  di- 

o'      i 

vided  by  the  consequent. 

11  1       1       1      b      b 

The  reciprocal  ratio,  is  —  'y  or  ~ T-J =7xT=~a ' 

ihat  is,  the  consequent  b  divided  by  the  antecedent  a. 

Hence  a  reciprocal  ratio  is  expressed  by  inverting  the  frac- 
tion which  expresses  the  direct  ratio  ;  or,  when  the  notatiow 
is  by  points,  by  inverting  the  order  of  the  terms. 

Thus  a  is  to  b,  inversely,  as  b  to  a. 

352.  COMPOUND  RATIO  is  the  ratio  of  the  PRODUCTS  of  the 
corresponding  terms  of  two  or  more  simple  ratios* 

'  Thus  the  ratio  of  6  : 3,  is  2 

And  the  ratio  of  12 :  4,  is  3 


^  The  ratio  compounded  of  these  is        72 : 12=6. 

Here  the  compound  ratio  is  obtained  by  multiplying  to- 
gether the  two  antecedents,  and  also  the  two  consequents,  of 
the  simple  ratios. 

So  the  ratio  compounded, 

Of  the  ratio  of  a  :  b 

And  the  ratio  of  c  :  d 

And  the  ratio  of  h  :  y 

ach 
Is  the  ratio  of  ach :  bdy=-r-f-  • 

Compound  ratio  is  not  different  in  its  nature  from  any  oth- 
er ratio.  The  term  is  used,  to  denote  the  origin  of  the  ra- 
tio, in  particular  cases. 

*See  note  D. 


RATIO.  179 

Cor.  The  compound  ratio  is  equal  to  the  product  of  the 
simple  ratios. 

The  ratio  of  a :  6.  is  -r- 

/    b 

The  ratio  ef  c:d,is~j 

h 
The  ratio  of  k:y,\s  — 

h 

And  the  ratio  compound  of  these  is       O~*  which  is  the 

product  of  the  fractions  expressing  the  simple  ratias.    (Art. 
Z55.) 

So  the  ratio  of  8  : 4  is  2 

The  ratio  of  6  : 2  is  3 

The  ratio  of  .  8  : 2  is  4 


And  the  ratio  compounded  of  these  is  24—2x3x4. 

353.  If,  in  a  series  of  ratios,  the  consequent  of  each  pre- 
ceding couplet,  is  the  antecedent  of  the  following  one,  the 
ratio  of  the  first  antecedent  to  the  last  consequent,  is  equal  to 
that  which  is  compounded  of  all  the  intervening  ratios* 

Thus,  in  the  series  of  ratios     a :  b 

b:c 
c  :d 
d:h 

the  ratio  of  a :  h  is  equal  to  that  which  is  compounded  of 
the  ratios  of  a  :  6,  of  6  :  c,  of  c :  d,  of  d ;  h.      For  the  com- 

abcd     a 
pound  ratio,  by  the  last  article,  is  17:71— T>  or  a :^«     (Art. 

145.) 

In  the  same  manner,  all  the  quantities  which  are  both  an- 
tecedents and  consequents  will  disappear  when  the  fractional 
product  is  reduced  to  its  lowest  terms,  and  will  leave  the  com- 
pound ratio  to  be  expressed  by  the  first  antecedent  and  the 
last  consequent. 

The  ratio  compounded  of  2 : 6 

6:8 
8: 15,  is  TyT=T*T  or  2:15.^' 

*  This  is  the  particular  case  of  compound  ratio  which  is  treated  of 
fa  the  5th  book  of  EueJid.  S*e  the  editions  of  Simson  and  Playfqjj-. 


ISO  ALGEBRA. 

354.  A  particular  class  of  compound  ratios  is  produced, 
by  multiplying  a  simple  ratio  into1  itself,  or  into  another  equal 
ratio.     These  are  termed  duplicate,  triplicate,  quadruplicate, 
&c.  according  to  the  number  of  multiplications. 

A  ratio  compounded  of  two  equal  ratios,  that  is,  the  square 
of  the  simple  ratio,  is  called  a  duplicate  ratio. 

One  compounded  of  three,  that  is,  the  cube  of  the  simple 
ratio,  is  called  triplicate,  Jkc. 

a 

Thus  the  simple  ratio  of  a  to*0,  is  a'-b  =x~r 

a* 
The  duplicate  ratio  of  a  to  b,  is  az  :  62  =77 

«3 
The  triplicate  ratio  of  a  to  b,  is  a3  :b3=-r^,  &c. 

The  terms  duplico/c,  triplicate,  &tc.  ought  not  to  he  con~ 
founded  with  double,  triple,  &c.* 

The  ratio  of  6  to  3  is  6:3=2 

Double  this  ratio,  that  is,  twice  the  ratio  is      12: 3=6  ) 
Triple  the  ratio,  i.  e.  three  times  the  ratio,  is  18:3=9) 

Butthe^t^Z^a£eratio,i.e.thes^wareoftheratio,is62  :22  =9  ) 
And  the  triplicate  ratio,i.e.the  cube  of  the  rations  6 3 : 2 3  =27  3 

355.  That  quantities  may  have  a  ratio  to  each  other,  it  is 
necessary  that  they  should  be  so  far  of  the  same  nature,  as 
that  one  can  properly  be  said  to  be  either  equal  to,  or  great- 
er, or  less  than  the  other.     A  foot  has  a  ratio  to  an  inch,  for 
one  is  twelve  times  as  great  as  the  other.     But  it  can  not  be 
said  that  an  hour  is  either  longer  or  shorter  than  a  rod  ;  or 
that  an  acre  is  greater  or  less  than  a  degree.     Still,  if  these 
quantities  are  expressed  by  numbers,  there  may  be  a  ratio 
between  the  numbers.     There  is  a  ratio  between  the  number 
of  minutes  in  an  hour,  and  the  number  of  rods  in  a  mile. 

356.  Having  attended  to  the  nature  of  ratios,  \ve  have 
next  to  consider  in  what  manner  they  will  be  affected,  by 
varying  one  or  both  of  the  terms  between  which  the  com- 
parison is  made.  It  must  be  kept  in  mind  that,  when  a  di- 
rect ratio  is  expressed  by  a  fraction,  the  antecedent  of  the 
couplet  is  always  the  numerator,  and  the  consequent,  the  de- 
nominator. It  will  be  easy,  then,  to  derive  from  the  proper- 
ties of  fractions,  the  changes  produced  in  ratios  by  varia- 
tions in  the  quantities  compared.  For  the  ratio  of  the  two 
quantities  is  the  same  as  the  value  of  the  fractions,  each  be* 

*  See  Note  E. 


RATIO.  181 

ing  the  quotient  of  the  numerator  divided  by  the  denomina- 
tor. (Arts.  135,  346.)  Now  it  has  been  shown,  (Art.  137.) 
that  multiplying  the  numerator  of  a  fraction  by  any  quantity 
is  multiplying  the  value  by  that  quantity ;  and  that  dividing 
the  numerator  is  dividing  the  value.  Hence, 

357.  Multiplying  the  antecedent  of  a  couplet  by  any  quanti- 
ty, is  multiplying  the  ratio  by  that  quantity  ;  and  dividing  the. 
antecedent  is  dividing  the  ratio. 

Thus  the  ratio  of     6  :  2  is  3 
And  the  ratio  of     24 : 2  is  12. 

Here  the  antecedent  and  the  ratio,  in  the  last  couplet,  are 
each  four  times  as  great  as  in  the  first. 

a 
The  ratio  of  a  :  b  is  y 

na 
And  the  ratio  of  na  :  b  is    y  • 

Cor.  With  a  given  consequent,  the  greater  the  antecedent, 
the  greater  the  ratio;  and  on  the  other  hand,  the  greater  the 
ratio,  the  greater  the  antecedent*  See  art.  137.  cor. 

358.  Multiplying  the  consequent  of  a  couplet  by  any  quan- 
tity is,  in  effect,  dividing  the  ratio  by  that  quantity;  and  divi- 
ding the  consequent  «  multiplying  the  ratio.      For  multiply- 
ing the  denominator  of  a  fraction,  is  dividing  the  value  ;  and 
.dividing  the  denominator  is  multiplying  the  value.    (Art.138.) 

Thus  the  ratio  of  12 : 2,  is  6 
And  the  ratio  of    12: 4,  is  3. 

Here  the  consequent,  in  the  second  couplet,  is  twice  as 
great,  and  the  ratio  only  halfzs,  great,  as  in  the  first. 

« 
The  ratio  of  a: b,  is  y 

a 

And  the  ratio  of  a :  nb.  is — >• 
no 

Cor.  With  a  given  antecedent,  the  greater  the  conse- 
quent, the  less  the  ratio ;  and  the  gi*eater  the  ratio,  the  less 
the  consequent.f  See  art.  138.  cor. 

359.  From  the  two  last  articles,  it  is  evident  that  multiply- 
ing the  antecedent  of  a  couplet,  by  any  quantity,  will  have  the 

*  Euclid  8  and  10.  b.     The  first  part  of  the  propositions. 
|  Euclid  8  and  10.  {?.    The  last  part  of  the  propositions. 


183  ALGEBRA. 

same  effect  on  the  ratio,  as  dividing  the  consequent,  by  that 
quantity;  and  dividing  the  antecedent  will  have  the  same  ef- 
iect  as  multiplying  the  consequent.  See  art.  139. 

Thus  the  ratio  of  8 : 4,  is  2 

Mult,  the  antecedent  by  2,  the  ratio  of    16:4,  is  4 
Divid.  the  consequent  by  2,  the  ratio  of    8:2,  is  4. 

Cor.  Any  factor  or  divisor  may  be  transferred,  from  the 
antecedent  of  a  couplet  to  the  consequent,  or  from  the  con- 
sequent to  the  antecedent,  without  altering  the  ratio. 

It  must  be  observed  that,  when  a  factor  is  thus  transferred 
from  one  term  to  the  other,  it  becomes  a  divisor ;  and  when 
a  divisor  is  transferred,  it  becomes  a  factor. 

Thus  the  ratio  of  3x6:9=2)     , 

Transferring  the  factor  8,          6  :  |  =2  5   the  same  rtttlo> 

win        ma  ma 

The  ratio  of  -  :&=—  +  b  =-r 

y       y          by 

•  f  ma 

Transferring  y,  ma :  by=ma-—by=-r-       f 

y 

by  by    ma 

Transferring  mt  « :  —  =  a  4-  —  =-?— 

»  m      by        J 

360.  It  is  farther  erideit,  from  arts.  357  and  358,  that  ?/ 
ihe  antecedent  and  consequent  be  BOTH  multiplied,  or  both  divi- 
ded, by  the  same  quantity,  the  ratio  ivill  not  be  altered.*     Sec  « 
art.  140. 

Thus  the  ratio  of  8 :  4=2  ^ 

Mult,  both  terms  by  2,    16  :  8=2  >  the  same  ratio. 

Divid.  both  terms  by  2,    4 :  2=2  ) 

The  ratio  of  «:&=-T- 


mn      a 


Mult,  bpth  terms  by  m,  ma:mb=~i=~r 

a  .  b      an      a 

Dmd.  both  terms  by  n,  — .— =r-=-r 

*  n    n     bn     b 

Cor.  1.  The  halves  of  quantities  have  the  same  ratio  as 
iheir  wholes. 

*  Euclid  15.  5. 


RATIO.  185 

Cor.  2.  The  ratio  of  two  fractions  which  hare  a  common 
denominator,  is  the  same  as  the  ratio  of  their  numerators. 

a     b 

The  ratio  of  —  :  — ,  is  the  same  as  that  of  a :  b. 
n     n' 

Cor.  3.  The  direct  ratio  of  two  fractions  which  have  a 
common  numerator,  is  the  same  as  the  reciprocal  ratio  of 
their  denominators. 

a     a  11 

Thus  the  ratio  of  —  :  — ,  is  the  same  as  —  :  —  or  n  :  m. 
m     n'  m     n 

361.  From  the  last  article,  it  will  be  easy  to  determine 
the  ratio  of  any  two  fractions.  If  each  term  be  multiplied 
by  the  two  denominators,  the  ratio  will  be  assigned  in  inte- 
gral expressions.  Thus,  multiplying  the  terms  of  the  coup- 

a     c  abd    bed 

let  T-  :  -y  by  bd,  we  have  -y-  :  -r  ,  which  becomes  ad :  be, 

by  cancelling  equal  quantities  from  the  numerators  and  de- 
nominators. 

3G2.  If  to  or  from  tJie  terms  of  any  couplet,  there  be  ADDED 
or  SUBTRACTED  tivo  other  quantities  having  the  same  ratio,  tht- 
sums  or  remainders  will  also  have  the  same  ratio.* 

Let  the  ratio  of  a :  b 


Be  the  same  as  that  of  c 


:&> 
:d\ 


Then  the  ratio  of  the  sum  of  the  antecedents,  to  the  sum 
•f  the  consequents,  viz.  of  a+c  to  b+d,  is  also  the  same. 

a+c      c      a 


Demonstration. 

a      c 

1.  By  supposjtion,  7~=~ff 

2.  Mult,  by  b  and  rf,  (Ax.  3.)  ad=bc 

3.  Adding  cd  to  both  sides,  (Ax.  1.)  ad+cd=bc+cd 

bc-\-cd 

4.  Dividing  by  d,  (Ax.  4.)  a+c  =  —  j  — 

a  +  c      c      a 

5.  Dividing  by  b+d,  (Art.  121.)  ==     ~ 


*Eudid5and6.  5.. 


184     ,  ALGEBRA. 

The  ratio  of  the  difference  of  the  antecedents,  to  the  dif- 
ference of  the  consequents,  is  also  the  same. 

.    a— c      c      a 
That  1ST ;:S=-T=:-J-' 

b— a     a      b 

Demonstration. 

a      c 

1.  By  supposition,  as  before,  T=7/ 

2.  Multiplying  by  b  and  d,  ad— be 

3.  Subtracting  cd  from  both  sides,         ad—cd=bc—cd 

bc—cd 

4.  Dividing  by  d,  a— c= — T — 

a—c      c      a 
/>.  Dividing  by  b—d  7 — ^=-r=-r* 

°  0  —~  (t       u        0 

Thus  the  ratio  of  15 : 5  is  3  ) 

And  the  ratio  of  9 : 3  is  3  5 

Then  adding  and  subtracting  the  terms  of  the  two  couplets. 

The  ratio  of  15  +  9  : 5  +  3  is  3} 

And  the  ratio  of  15— 9:5— 3  is  3  5 

Here  the  terms  of  only  two  couplets  have  been  added  to- 
gether. But  the  proof  may  be  extended  to  any  number  of 
couplets,  where  the  ratios  are  equal.  For,  by  the  addition 
of  the  two  first,  a  new  couplet  is  formed,  to  which,  upon  the 
same  principle,  a  third  may  be  added,  a  fourth,  &c.  Hence, 

363.  If,  in  several  couplets,  the  ratios  are  equal,  the  sum 
of  all  the  antecedents  has  the  same  ratio  to  the  sum  of  all  the 
consequents,  which  any  one  of  the  antecedents  has,  to  its  conse- 
quent.* 

f  12: 6=2 

I     *  I  C 

Thus  the  ratio  of  <    fi  i^Ho 
L  6:3=2 
Therefore  the  ratio  of  (12+10  +  84-6):  (6 +5-f  4+3)=Z 

*  Euclid  1  and  12.  5. 


PROPORTION.  185 

PROPORTION. 

363.  An  accurate  and  familiar  acquaintance  with  the  doc- 
trine of  ratios,  is  necessary  to  a  ready  understanding  of  the 
principles  of  proportion,  one  of  the  most  important  of  all 
the  branches  of  the  mathematics.     In  considering  ratios,  we 
compare  two  quantities,  for  the  purpose  of  finding  either 
their  difference,  or  the  quotient  of  the  one  divided  by  the 
other.      But  in  proportion,  the  comparison  is  between  two 
ratios.     And  this  comparison  is  limited  to  such  ratios  as  are 
equal.     We  do  not  inquire  how  much  one  ratio  is  greater  or 
less  than  another,  but  whether  they  are  the  same.      Thus  the 
numbers  12,  G,  8,  4,  are  said  to  be  proportional,  because  the 
ratio  of  12:6  is  the  same  as  that  of  8:4. 

364.  PROPORTION,  then,  is  an  equality  of  ratios.     It  is  ei- 
ther arithmetical  or  geometrical.     Arithmetical  proportion  is 
an  equality  of  arithmetical  ratios,  and  geometrical  propor- 
tion is  an  equality  of  geometrical  ratios.*      Thus  the  num- 
bers 6,  4,  10,  8,  are  in  arithmetical  proportion,  because  the 
difference  between  6  and  4  is  the  same  as  the  difference  be- 
tween 10  and  8.      And  the  numbers  6,  2,  12,  4,  are  in  geo- 
metrical proportion,  because  the  quotient  of  6  divided  by  2  is 
the  same,  as  the  quotient  of  12  divided  by  4. 

365.  Care  must  be  taken  not  to  confound  proportion  with 
ratio.     This  caution  is  the  more  necessary,  as  in  common 
discourse,  the  two  terms  are  used  indiscriminately,  or  rather, 
proportion  is  used  for  both.     The  expenses  of  one  man  are 
said  to  bear  a  greater  proportion  to  his  income,  than  those 
of  another.     But  according  to  the  definition  which  has  just 
been  given,  one  proportion  is  neither  greater  nor  less  than 
another.   For  equality  does  not  admit  of  degrees.    One  ratio 
may  be  greater  or  less  than  another.      The  ratio  of  12:2  is 
greater  than  that  of  6:2,  and  less  than  that  of  20 : 2.     But 
these  differences  are  not  applicable  to  proportion,  when  the 
term  is  used  in  its  technical  sense.      The  loose  signification 
which  is  so  frequently  attached  to  this  word  may  be  proper 
enough  in  familiar  language.     For  it  is  sanctioned  by  gene- 
ral usage.      But,  for  scientific  purposes,  the  distinction  be- 
tween proportion  and  ratio,  should  be  clearly  drawn,  and  cau- 
tiously observed. 

*  See  Note  F. 


ISG  ALGEBRA. 

366.  The  equality  between  two  ratios,  as  has  been  stated, 
is  called  proportion.     The  word  is  sometimes  applied  also  to 
the  series  of  terms  among  which  this  equality  of  ratios  ex- 
ists.    Thus  the  two  couplets  15:5  and  6 : 2  are,  when  taken 
together,  called  a  proportion. 

367.  Proportion  may  be  expressed,  either  by  the  common 
*5gn  of  equality,  or  by  four  points  between  the  two  couplets. 

Th     $  ®  "  6  ~^  "  2,  or  8  ••  6 :  :4  ••  2  )     are  arilhmeticaF 
(  «  ••  b  =  c  ••  d,  or  a  -b  : :c  ••  d  )       proportions. 

.    .   (  12:6=8:4,  or  I2:6::8:4)       are  geometrical 

\    a  :  b=d :  A,  or    a :  b  : :  d: h  )          proportions. 

The  latter  is  read,  '  the  ratio  of  a  to  b  equals  the  ratio  of 
d  to  A;1  or  more  concisely,  'a  is  ta  b,  as  d  to  A.' 

368.  The  first  and  last  terms  are  called  the  extremes,  and 
the  other  two  the  means-.     Homologous  terms  are  either  the 
two  antecedents  or  the  two  consequents.     Analogous  terms 
are  the  antecedent  and  consequent  of  the  same  couplet. 

369.  As  the  ratios  are  equal,  it  is  manifestly  immaterial 
which  of  the  two  couplets  is  placed  first. 

a      e          c      a 
If  a:b*:c  :-a,  then  c  :d::a  :  b.     For  n  -7-^=-Tthen--T=-T-. 

370.  The  number  of  terms  must  be,  at  least,  four.     For 
the  equality  is  between  the  ratios  of  two  couplets  ;  and  each 
couplet  must  have  an  antecedent  and  a  consequent.     There 
may  be  a  proportion,  however,  among  three  quantities.     For 
one  of  the  quantities  may  be  repeated,  so  as  to  form  two 
terms.     In  this  case,  the  quantity  repeated  is  called  the  mid- 
dle term,  or  a  mean  proportional  between  the  two  other  quan- 
tities, especially  if  the  proportion  is  geometrical. 

Thus  the  numbers  8,  4,  2,  are  proportional.  That  is, 
8  : 4:  :4 : 2.  Here  4  is  both  the  consequent  in  the  first  coup- 
let, and  the  antecedent  in  the  last.  It  is  therefore  a  mean 
proportional  between  8  and  2. 

The  last  term  is  called  a  third  proportional  to  the  two  oth- 
er quantities.  Thus  2  is  a  third  proportional  to  8  and  4. 

371.  Inverse  or  reciprocal  proportion  is  an  equality  between 
a  direct  ratio  and  a  reciprocal  ratio. 

Thus  4  :-2::-j- :  -*  ;  that  is,  4  is  to  2,  reciprocally,  as  3  to  6. 
Sometimes  also,  the  order  of  the  terms  in  one  of  the  coup- 
lets is  inverted,  without  writing,  them  in  the  fb^m  of  a  frac- 
iion.  (Art.  351.) 


PROPORTION. 

Thus  4 : 2: :3 : 6  inversely.  In  this  case,  the  first  term  is 
1.0  the  second,  as  the  fourth  to  the  third  ;  that  is,  the  first  di- 
vided by  the  second,  is  equal  to  the  fourth  divided  by  the 
third. 

372.  When  there  is  a  series  of  quantities,  such  that  the  ra- 
tios of  the  first  to  the  second,  of  the  second  to  the  third,  of 
the  third  to  the  fourth,  kc.  are  all  equal ;  the  quantities  are 
said  to  be  in  continued  proportion.     The  consequent  of  each 
preceding  ratio  is,  then,  the  antecedent  of  the  following  one. 
Continued  proportion  is  also  called  progression,  as  will  be 
seen  in  a  following  section. 

Thus  the  numbers  10,  8,  6,  4,  2,  are  in  continued  arithme- 
tical proportion.  For  10  —  8=8-6=6—4=4—2. 

The  numbers  64,  32,  16,  8, 4,  are  in  continued  geometrical 
proportion.  For  64  : 32::  32:16  ::  16:8  :: 8:4. 

If  a,  ft,  e,  d,  A,  &c.  are  in  continued  geometrical  propor- 
tion j  then  .a: 6  ::b:c::c:d::d:h,  &c. 

One  case  of  continued  proportion  is  that  of  three  propor- 
tional quantitlties.  (Art.  370.) 

373.  As  arithmetical  proportion  is,  generally,  nothing  more 
than  a  very  simple  equation,  it  is  scarcely  necessary  to  treat 
of  it  separately. 

The  proportion  a  ••  b  : :  c  ••  d 

Is  the  same  as  the  equation  a—b—c—d. 

It  will  be  proper,  however,  to  observe  that,  if  four  quan- 
tities are  in  arithmetical  proportion,  the  sum  of  the  extremes 
is  equal  to  the  sum  of  the  means. 

Thus  if  a  ~b::h  "in, then  a+m=b-{-h 

For  by  supposition,  a—b—h  —  m 

And  transp.  —  b  and  —  m  a+m=b-{-h 

Sointheproportion,12-10::ll  ••  9,we have  12+9  =  10+11. 

Again,  if  three  quantities  are  in  arithmetical  proportion, 
the  sum  of  the  extremes  is  equal  to  double  the  mean. 

If  a  -  b  ::b  ••  c,  then  a  — b=b— « 

And  transposing  —b  and  —  c, 


GEOMETRICAL   PROPORTION. 

374.  But  if  four  quantities  are  in  geometrical  proportion,  the 
PRODUCT  of  the  extreme*  is  equal  to  the  product  of  the  means. 


188  ALGEBRA. 

If   a:b  ::c:ef,  then  ad— be 

a      c 
For  by  supposition,  (Arts.  346,  364.)         T~~d 

abd     cbd 
Multiplying  by  Id,  (Ax.  3.)  ~r~  =~j~ 

Reducing  the  fractions,  ad=bc. 

Thus  12:  8::  15: 10,  therefore  12x10=8x15. 

375.  On  the  other  hand,  if  the  product  of  two  quantities 
is  equal  to  the   product  of  two   others,  the  four  quantities 
will  form  a  proportion,  when  ihey  are  so  arranged,  that  those 
on  one  side  of  the  equation  shall  constitute  the   means,  and 
those  on  the  other  side,  the  extremes. 

jn 
If  my=nh,  then  m:n::h:y,  that  is       --: 

my    nh 
For  by  dividing  my=nh  by  ny,  we  have  —  = — 

, 
m      n 

And  reducing  the  fractions,  — = — 

tt      y 

Cor.  The  same  must  be  true  of  any  factors  which  form 
the  two  sides  of  an  equation. 

If  (a  +  b]  xc  =  (d—m)xy,  then  a+b  :d—m  ::y  :c. 

376.  If  three  quantities  are  proportional,  the  product  of 
the  extremes  is  equal  to^lhe  square  of  the  means.     For  this 
mean  proportional  is,  at  the  same  time,  the  consequent  of 
the  first  couplet,  and  the  antecedent  of  the  last.  (Art.  370.)  It 
is  therefore  to   be  multiplied  into  itself,  that  is,  it  is  to  be 
squared. 

If  a:b::b:c,  then  mult,  extremes  and  means,  ac=b2. 

377.  It  follows  from  art.  374,  that  in  a  proportion,  either 
extreme  is  equal  to  the  product  of  the  means,  divided  by 
the  other  extreme  ;  and  either  of  the  means  is  equal  to  the 
product  of  the  extremes,  divided  by  the  other  mean. 

•>•'/''. -?f ..+  /' 


PROPORTION.  189 

1.  If  a:b  ::c:d,  then  ad=bc 

2.  Dividing  by  d,  a~~d 

ad 

3.  Dividing  the  first  by  c,     b—— 

c 

,  ad 

4.  Dividing  it  by  b,  C=7T 

be 

5.  Dividing  it  by  a,  d——  ;  that  is,  the  fourth  term  is 

equal  to  the  product  of  the  second  and  third  divided  by  the  first. 

On  this  principle  is  founded  the  rule  of  simple  proportion 
in  arithmetic,  commonly  called  the  Rule  of  Three.  Three 
numbers  are  given  to  find  a  fourth,  which  is  obtained  by 
multiplying  together  the  second  and  third,  and  dividing  by 
the  first. 

378.  The  propositions  respecting  the  products  of  the 
means,  and  of  the  extremes,  furnish  a  very  simple  and  con- 
venient criterion  for  determining  whether  any  four  quanti- 
ties are  proportional.  We  have  only  to  multiply  the  means 
together,  and  also  the  extremes.  If  the  two  products  arc 
equal,  the  quantities  are  proportional.  If  the  products  are 
not  equal,  the  quantities  are  not  proportional. 

379.  In  mathematical  investigations,  when  the  relations  of 
several  quantities  are  given,  they  are  frequently  stated  in  the 
form  of  a  proportion.  But  it  is  commonly  necessary  that 
this  first  proportion  should  pass  through  a  number  of  trans- 
formations, before  it  brings  out  distinctly  the  unknown  quan- 
tity, or  the  proposition  which  we  wish  to  demonstrate.  It 
may  undergo  any  change  which  will  not  affect  the  equality 
of  the  ratios;  or  which  will  leave  the  product  of  the  means 
equal  to  the  product  of  the  extremes. 

It  is  evident,  in  the  first  place,  that  any  alteration  in  the 
arrangement,  which  will  not  affect  the  equality  of  these  two 
products,  will  not  destroy  the  proportion.  Thus,  if  a  :  6  : :  c:d, 
the  order  of  these  four  quantities  may  be  varied,  in  any  way 
which  will  leave  ad=bc.  Hence, 

380.  If  four  quantities  are  proportional,  the  order  of  the 
means,  or  of  the  extremes,  or  of  the  terms  of  both  couplets, 
may  be  inverted,  trithout  destroying  the  proportion. 


190  ALGEBRA. 

If          a  :  1) : :  c  :  d 


then, 


And  12  :  8  : :  6  : 

1.  Intcrting  the  means,* 

a  :  c :  •  I :  d  )     ,       .     (  Tlie  first,  is  to  the  third, 

12:6::8:4  V  '  /  As  the  second,  to  the  fourth. 

'  ^  • 

In  other  words,  the  ratio  of  the  antecedents  is  equal  to  the 
ratio  of  the  consequents. 

This  inversion  of  the  means  is  frequently  referred  to  by 
geometers  under  the  name  of  Alternation.^ 

. 

2.  Inverting  the  extremes. 

d:b::c:a    >,    .  •     (  The  fourth,  is  to  the  second, 
4 : 8 : :  6 : 12  5  ll      IS'  {  As  the  third,  to  the  jinL 

v 
J.  Inverting  the  terms  of  each  couplet. 

b  :  a  ::d:c")    ,    .  «     C  The  second,  is  to  the  first, 
8 : 1 2 : :  4 : 6  $  l       IS'  £  As  the  fourth,  to  the  *Am/. 
This  is  technically  called  Inversion. 

Each  of  these  may  also  be  varied,  by  changing  the  order 
of  the  two  couplets.  (Art.  369.) 

Cor.  The  order  of  the  whole  proportion  may  be  inverted. 

If  a:b::c:d,  then  d :  c  ::  b  :  a. 

In  each  of  these  cases,  it  will  be  at  once  seen  that,  by 
taking  the  products  of  the  means,  and  of  the  extremes,  we 
have  ad— be,  and  12x4=8x6. 

If  the  terms  of  only  one  of  the  couplets  are  inverted,  the 
proportion  becomes  reciprocal.  (Art.  371.) 

If  a  :  b : :  c  :  d,  then  a  is  to  b,  reciprocally  as  d  to  c. 

381.  A  difference  of  arrangement  is  not  the  only  alteration, 
which  we  have  occasion  to  produce,  in  the  terms  of  a  pro- 
portion. It  is  frequently  necessaiy  to  multiply,  divide,  in- 
Yolve,  &c.  In  all  cases,  the  art  of  conducting  the  investiga- 
tion consists  in  so  ordering  the  several  changes,  as  to  main- 
tain a  constant  equality,  between  the  ratio  of  the  two  first 
terms,  and  that  of  the  two  last.  As  in  resolving  an  equa- 
tion, we  must  see  that  the  sides  remain  equal ;  so  in  varying 
a  proportion,  the  equality  of  the  ratios  must  be  preserved. 
And  this  is  effected,  either  by  keeping  the  ratios  the  same, 

*  See  Note  G.  t  Euclid  16.  5. 


PROPORTION.  191 

while  the  terms  are  altered  ;  or  by  increasing  or  diminishing- 
one  of  the  ratios,  as  much  as  the  other.  Most  of  the  succeed- 
ing proofs  are  intended  to  bring  this  principle  distinctly  into 
view,  and  to  make  it  familiar.  Some  of  the  proposition* 
might  be  demonstrated,  in  a  more  simple  manner,  perhaps, 
by  multiplying  the  extremes  and  means.  But  this  would  not 
give  so  clear  a  view  of  the  nature  of  the  several  changes  in 
the  proportions. 

It  has  been  shown  that,  if  both  the  terms  of  a  couplet  be  mul- 
tiplied or  divided  by  the  same  quantity,  the  ratio  will  remain 
the  same ;  (Art.  360.)  that  multiplying  the  antecedent  is,  in 
effect,  multiplying  the  ratio,  and  dividing  the  antecedent  is 
dividing  the  ratio ;  (Art.  357.)  and  farther,  that  multiplying 
the  consequent  is,  in  effect,  dividing  the  ratio,  and  dividing 
the  consequent  is  multiplying  the  ratio.  (Art.  358.)  As  the 
ratios  in  a  proportion  are  equal,  if  they  are  both  multiplied, 
or  both  divided,  by  the  same  quantity,  they  will  still  be  equal. 
(Ax.  3.)  One  will  be  increased  or  diminished  as  much  as 
the  other.  Hence, 

382.  If  four  quantities  are  proportional,  two  analogous,  or 
two  homologous  terms  may  be  multiplied  or  divided  by  the  same 
Quantity,  without  destroying  the  proportion. 

If  analogous  terms  be  multiplied  or  divided,  the  ratios  will 
not  be  altered.  (Art.  360.)  If  homologous  terms  be  multi- 
plied or  divided,  both  ratios  will  be  equally  increased  or  di- 
minished. (Arts.  357:  8.) 


If         a:Jt::c:d^   , 

And  1S:8::6:45 

. 

1.  Multiplying  ike  two  Jirst  terms, 

ma  :  mb   :  :  c  :  d 
2  X  12  :2  x  8  :  •  6  •  4 


)  m,         *•  u      j 

S  °S         a"ered- 


. 

Multiplying  the  two  last  terms. 

a:b  ::  me   :  md      )  r™        ,.          .    u       i 
12:8::2x6:2X4TheratlOSn0taltered- 

3.  Multiplying  the  two  antecedents.* 
2  x*12  '  8  •  •  2xf)  '•  1    S  ^0*  ratl°S 

'+  Euclid  S.  5. 


192-  ALGEBRA. 

4.  Multiplying  the  two  consequents. 

a:  mb   : :  c :  mil     ?  T>  .1       »•  n     j-    •  •  u  j 

12-2x8::  a:  2x4S  equally  diminished. 

5.  Dividing  the  two  first  terms. 

a      b 

—  :  —  : :  c :  d.     The  ratios  not  altered. 
in    m 

i.  Dividing  the  two  last  terms. 

.      c      d  . 

a  :  6  : :  —  :  —     The  ratios  not  altered. 

/n     m 

T.  Dividing  the  two  antecedents. 

a          c 

—  :b::—:d.     The  ratios  equally  diminished. 
m         m  i      j 

9.  Dividing  the  two  consequents. 

b          d 

a :  ~  : :  c :  —     The  ratios  equally  increased. 

mm  *      J 

Cor.  1.  All  the  terms  may  be  multiplied  or  divided  b* 
the  same  quantity.* 


ma  :mb::mc:  md  } 
a     b       c      d      > 

m  '  m  "  m  '  m     / 


The  ratios  not  altered. 


Cor.  2.  If  four  quantities  are  proportional,  their  halves- 
are  proportional. 

Cor.  3.  In  any  of  the  cases  in  this  article,  multiplication 
of  the  consequent  may  be  substituted  for  division  of  the  an- 
tecedent in  the  same  couplet,  and  division  of  the  conse- 
quent, for  multiplication  of  the  antecedent.  (Art.  359,  cor.) 

f  1  *   f      *  ,1        f  d 

\ma:b::mc:d  {   °*  }  a:     ::mc:d  \        \ma'.b::c:  — 

ThusJ  C^J      »  »' 

for    )  «    ,      c  ^  c 

U:J::m:jJ  § 

*  Euclid  4.  5. 


PROPORTION.  193 

363.  h  is  often  necessary,  not  only  to  alter  the  terms 
of  a  proportion,  and  to  vary  the  arrangement,  but  to  compare 
one  proportion  tvith  another.  From  this  comparison  will  frer- 
quently  arise  a  new  proportion,  which  may  be  requisite  in 
solving  a  problem,  or  in  carrying  forward  a  demonstration. 
One  of  the  most  important  cases  is  that  in  which  two  of  the 
terms  in  one  of  the  proportions  compared,  are  the  same  with 
two  in  the  other.  The  similar  terms  may  be  made  to  disap- 
pear, and  a  new  proportion  may  be  formed  of  the  four  re- 
maining terms.  For, 

384.  If  t^uo  ratios  are  respectively  equal  to  a  third,  they  are 
equal  to  each  other* 

This  is  nothing  more  than  the  llth  axiom  applied  to 
ratios. 

1.  If    a: b  ::m:n>  .,  7         j  77/4  *.non\ 
Andc:d::m:nyh™a:b::c:d,ora:c::b:d.(tet.3QQ.} 

2.  If      a  :  b  : :  m  :  n  >  .,  7  7  7     T 
*    j                     i  >  then  a :  b  : : c :  d.  or  a  :  c : :  b  :  d, 

Aim  in  '.n  '.'.  c  '.ay 

Cor.  If  a :  b  : :  m  :  n  >  .,  7 

™.^,.^thena:6>c:<7't 


For  if  the  ratio  of  771 :  n  is  greater  tho,n  that  of  c :  d,  it  is 
manifest  that  the  ratio  of  a :  b,  which  is  equal  to  that  of  m :  n, 
is  also  greater  than  that  of  c :  d. 

385.  In  these  instances,  the  terms  which  are  alike,  in  the 
two  proportions,  are  the  two  first  and  the  two  last.  But  this 
arrangement  is  not  essential.  The  order  of  the  terms  may 
be  changed,  in  various  ways,  without  affecting  the  equality  of 
the  ratios. 

1.  The  similar  terms  may  be  the  two  antecedents,  or  the 
two  consequents,  in  each  proportion.  Thus, 

, 

If      m  :  a : :  n  :  b  >  ^^  <  By  alternation,  m:n::a:l 

And  m, :  c  : :  n :  d  $  (  And  m  :  n  : :  e :  d. 

Therefore  a  :  b  : :  c :  d,  or  «  :  c  : :  b  :  d,  by  the  last  article, 

Or  if  a  :  m : :  b  :  n  >    j        (  By  alternation,  a:l::m:n 
And    c:m::d:n)  (And  «:d::m:n* 

Therefore  a :  I : :  c  :  d,  or  a :  e : :  b :  d. 

*  Euclid  11.  5.  f  Euclid  li,  5, 

A  a 


194  ALGEBRA. 

2.  The  antecedents  in  one  of  the  proportions,  may  be  the 
same  as  the  consequents  in  the  other. 

If     m:  a::  n  :  b  >    ,        (  By  invcr.  and  altern.  a:li:m:n 
And  c  :  m  ::</:»)  ^  By  alternation,          c  :  <?  :  :  m  :  ?» 

Therefore  a  :  6,  Jkc.  as  before. 

Or  if  a  :  m  :  :  b  :  n  )    .        <  By  alternation,  fl  :  ft  :  :  m:  n 

And  m  :  c  :  :  71:  c?  \  a  (  By  inver.  and  ahern.  c  :  «?  :  :  m  :  TV 
Therefore  a:  b,  &,c. 

3.  Two  homologous  terms,  in  one  of  the  proportions,  may 
be  the  same,  as  two  analogous  terms  in  the  other. 

If      a:  m  ::&:»)   *        ^  By  alternation,     a:b::m:n 

And  c  :  d  :  :  m  :  n  )  (  And  c  :  d  :  :  m  :  n 

Therefore  a  :  b,  &c. 

Or  if  a  :  b  :  :  m  :  n  )   ,        (  The  first  is  a:b::m:n 

And    c  :  m  :  :  d  :  n  )  {Ity  alternation,     c  :  d  :  :  j»  :  n 

on         r          c 
Therefore,  &c. 

All  these  are  instances  of  an  equality,  between  the  ratios' 
in  one  proportion,  and  those  in  another.  In  geometry,  the 
proposition  to  which  they  belong  is  usually  cited  by  the 
words  "  ex  aequo"  or  "  ex  aequali.tf*  The  second  case  i» 
this  article  is  that  which,  in  its  form,  most  obviously  answers 
to  the  explanation  in  Euclid.  But  they  are  all  upon  the 
same  principle,  and  are  frequently  referred  to,  without  dis- 
crimination. 

386.  Any  number  of  proportions  may  be  compared,  in 
the  ''same  manner,  if  the  two  first  or  the  two  last  terms  in 
each  preceding  proportion,  are  the  same  with  the  two  first 
or  the  two  last  in  the  following  one.* 

Thus  if  a  :  b  :  :  c  :  d 


A 

And       h  :  I  :  :  m  :  n  f 

And      m:n::x:yj 

That  is,  the  two  first  terms  of  the  first  proportion-  have  th« 
&une  ratio,  as  the  two  last  terms  of  the  last  proportion.  For 
it  is  manifest  that  tLe  ratio  of  all  the  couplets  is  the  same. 

And  if  the  terms  do  not  stand  in  the  same  order  as  here, 

*  Euclid  22.  5. 


PROPORTION.  135 

yet  If  they  can  be  reduced  to  this  form,  the  same  principle  is 
applicable. 

Thus  if  a  :  c  : :  b  :  d 

And    c:  h  ::d:l  \  .,       ,  .. 

A  7       > then  by  alternation- 

And   h:m::l:n  \ 

And  m :  x 

Therefore  a :  b ::  x :  y,  as  before. 

In  all  the  examples  in  this,  and  the  preceding  articles,  the 
two  terms  in  one  proportion  which  have  equals  in  another, 
are  neither  the  two  means,  nor  the  two  extremes,  but  one  of 
the  means,  and  one  of  the  extremes;  and  the  resulting  pro- 
portion is  uniformly  direct. 

387.  But  if  the  two  means,  or  the  two  extremes,  in  one 
proportion,  be  the  same  with  the  means,  or  the  extremes,  in 
another,  the  four  remaining  terms  will  be  reciprocally  pro- 
portional. 

.rft    < 

If      a:m::n:S)  1      1 

And  C'.mnnid]  then  a:  c::  j  :  ^,  or  a:  c::  d:b. 

For 
And  cd- 

In  this  example,  the  two  means  in  one  proportion,  are  like 
those  in  the  other.  But  the  principle  will  be  the  same,  if 
the  extremes  are  alike,  or  if  the  extremes  in  one  proportion 
are  like  the  means  in  the  other. 

K.I   .  /  .  i  ") 
lit  9il..O.nf.i  t     f 

A    ,  7       >  then  a:c::d:b. 

And  m : c : : a : n  ) 

Or  if  a :  m : :  n :  b  >  ,,  ,   7 

A  j      f  then  a:c::d*.b. 

And  m:  c  ::a:n  ) 

The  proposition  in  geometry  which  applies  to  this  case  is 
usually  cited  by  the  words  "  ex  aequo  perturbate"* 

388.  Another  way  in  which  the  terms  .of  a  proportion  may 
be  varied,  is  by  addition  or  subtraction. 

If  to  or  from  two  analagous  or  two  homologous  terms  of  ft 
proportion,  two  other  quantities  having  the  same  ratio  be  added 
or  subtracted,  the  proportion  will  be  preserved.^ 

For  a  ratio  is  not  altered,  by  adding  to  it,  or  subtracting 
from  it,  the  terras  of  another  equal  ratio.  (Art.  362.) 

*  Euclid  23.  5.  f  Euclid  4.  §. 


=mn  >  ^  3?4  .  Thercfore  ai=cd  and  a  .  c . . 
=wm  )  v 


10$  ALGEBRA. 

If        n  :  It : :  c  :  d 
And   a:  b::  m:  n 

Then  by  adding  to,  or  subtracting  from  a  and  b,  the  terms 
of  the  equal  ratio  m :  n,  we  have, 

a+w-b+n  ::  c:d  and  «  — m:b— n;:c:d. 

And  by  adding  and  subtracting  m  and  n,  to  and  from  c  and 
d,  we  have, 

a:b  ::c+m  :d-\-n,  and  a  :b::c—m  :  d—n. 

Hero  the  addition  and  subtraction  are  to  and  from  annlv- 
gous  tewns.  I  Jut  by  alternation,  (Art.  380.)  these  terms 
t\ill  become  homologous,  and  we  shall  have, 

a-\-m:  c  ::  b  +  n:  d,  and  «— m:c  ::l>  —  n:d. 

Cor.  1.  This  addition  may,  evidently,  be  extended  to  any 
number,  of  equal  ratios.* 

C  c:d 

Tims  if  a:b:J  h:l 
|  m:  n 

Then  a  :b::c+k-\-m+x:d+l+n+ y. 

Cor.  2.  If  a:ft::c:<O  .,  7 

A          .*.      .  i>  then  a+m:0::c+n:a.t 
^vna  m.o..re.a_) 

For  by  alternation  a : c : :  b :  d  >  there-  (      «  +  m :  c  +  n  : :  b :  d 
And  m:n::b:d)    fore    \  ora+m  :b  ::  c+n  :  d. 

389.  From  the  last  article  it  is  evident  that  if,  in  any  pro- 
portion, the  terms  be  added  to,  or  subtracted  from  each  oth- 
er, that  is, 

Tf  two  analogous  or  homologous  terms  be  added  to,  or  sub- 
tracted from  the  two  others,  the  proportion  will  be  preserved. 

Thus  if  a :  b  : :  c  :  d,        and  12  :  4  : :  6  :  2,  then, 

u 
I .  Adding  the  two  las.1  terms,  to  the  two  first. 

'  a+c:b+d::a:b  12  +  6  :4+2.'fl2:4 

and  a  +  c:b+d::c:d  12+6:4+2::  6  :2 

or  a+c:a::b  +  d:b  12+6  : 12.-.-4+ 2:4 

and  a+c:c::b  +  d:d.  J2-f6:    6::4-|-2;2, 

*  Euclid  2.  5.  Cor. 

. 

,f  Euclid  24.  5. 


PROPORTION.  197 

2.  Adding  the  two  antecedents,  to  the  two  consequents. 
a+b:b::c+d:d  12+4:  4  :;6  +  2:2 

a  +  b:a:;c-t  d  :  c,  &c.          12+4:12::6  +  2:6,  &c. 
This  is  called  Composition.* 

3.  Subtracting  the  two  ^/irsf  terms,  from  the  two  Zasf. 

c  —  a:  a  ::<?—&  :  b 
c—a  :  c  ::  d—b  :  d,  &c. 

4.  Subtracting  the  two  Z«s£  terms,  from  the  two  first. 

a—c  :b  —  d::a  :  if 
a—  c  :  b  —  d  ::  c  :  d,  &c. 

5.  Subtracting  the  consequents,  from  the  antecedents. 

a  —  b  :  b  ::  c  —  d  :  d 
a  :  a—b  ::  c  :  c—d,  &:c. 

The  alteration  expressed  by  the  last  of  these  forms  is  cal- 
led Conversion. 

G.  Subtracting  the  antecedents,  from  the  consequents. 
b—a  :  a  ::  d—c:c 
b  :  b  —  a  ::  d  :  d—c,  &tc. 


7.  Adding  and  subtracting. 


a+b  :  a—b  ::a+d:  a—d. 

,, 

That  is,  the  sum  of  the  two  first  terms,  is  to  their  differ- 
ence, as  the  sum  of  the  two  last,  to  their  difference. 

Cor.  If  any  compound  quantities,  arranged  as  in  the  pre- 
ceding examples,  are  proportional,  the  simple  quantities  of 
which  they  are  compounded  are  proportional  also. 

Thus  if  a-f  b  :  b  :  :  c+d  :  d,  then  a  r  b  :  :  c  :  d. 

This  is  called  Division.^. 

390.  If  the  corresponding  terms  of  two  or  more  ranks  of 
proportional  quantities  be  MULTIPLIED  together,  the  products 
will  be  proportional. 

This  is  compounding  ratios,  (Art.  352.)  or  compounding 
proportions.  It  should  be  distinguished  from  what  is  called 
romposition,  which  is  an  addition  of  the  terms  of  a  ratio. 
(Art.  389.  2.) 

*  Euclid  18.  5.  f  Euclid  19.  5. 

}  Euclid  17.  5.    See  Note  H. 


ifcg  ALGEBRA. 

If      a:b::6:d->  12:4::G:2 


And  h  : I : :  m :  n  \  10 


:4::G:2> 

:  5  : :  S  :  4  $ 


Then       «A  :  W  :  :  cm  :  rfn.  1  20  :  20  ;  :  48  :  8. 

For,  from  the  nature  of  proportion,  the  two  ratios  in  the 
first  rank  are  equal,  and  also  the  ratios  in  the  second  rank. 
And  multiplying  the  corresponding  terms  is  multiplying  the 
ratios,  (Art.  352.  cor.)  that  is,  multiplying  equals  by  equals  ; 
(Ax.  3.)  so  that  the  ratios  will  still  be  equal,  and  therefore 
the  four  products  must  be  proportional. 

The  same  proof  is  applicable  to  any  number  of  propor- 
tions. 


a:::c: 

If  <  h  :  I  :  :  m  :  n 
j 
(p:q::x:y 

Then  ahp  :  bin  :  :  cmx  :  dny. 

From  this  it  is  evident,  that  if  the  terms  of  a  proportion 
be  multiplied,  each  into  itself,  that  is,  if  they  be  raised  to 
any  power,  they  will  still  be  proportional. 

l{a:b::c:d  2:4::6:12 

a:b::c:d  2:4::  6:  12 


Then  a2  :  6*  ::  c2  :  d*  4:16 : :  36 : 144 

Proportionals  will  also  be  obtained,  by  reversing  this  pro- 
cess, that  is,  by  extracting  the  roots  of  the  terras. 

If  a :  b  ::c :  <7,  then  \/a  :  Jb  : :  v/c  :  Jd. 

For,  taking  the  product  of  ext.  and  means,  ad=bc 
And  extracting  both  sides,  ^/ad=^/bc 

That  is,  (Arts.  259,  375.)  Ja:jb;:Jc  :  Jd. 

Hence, 

391.  If  several  quantities  are  proportional,  their  like  pow- 
ers or  like  roots  are  proportional* 

If  a  :b  ::c:d 

' 

*  It  must  not  be  inferred  from  this,  that  quantities  have  the  same 
ratio,  as  their  like  powers  or  like  roots.    See  art.  354. 


PROPORTION. 

Then  an :  bn : :  c" :  d  n.  and  m/cr :  m^b  : :  *ye  :  mjd. 

And  "Van : m •/ i"  : :  mjcn :  m^d* ,  that  is,  a» :  6* : :  c» :  rf» . 

392.  If  the  terms  in  one  rank  of  proportionals  be  divided 
Ity  the  corresponding  terms  in  another  rank,  the  quotients 
will  be  proportional. 
This  is  sometimes  called  the  resolution  of  ratios. 

If      a:  b  ::c   :<?)  12:  6  ::  18:9) 

And  h:  I  ::m  :n  J  6:2  .-:    9:3$ 

J^A      _i.^  1?    A..1?    J?_ 

This  is  merely  reversing  the  process  in  art.  390,  and  may 
be  demonstrated  in  a  similar  manner : 

Or  thust 

Taking  the  product  of  ext.  and  means,     ad=bc  > 
And  /m=Zm> 

ad     be 
Dividing  one  by  the  other,  -r-=j— 

That  is,  (Art.  375.)  l:J::^:«r 


This  should  be  distinguished  from  what  geometers  call 
division,  which  is  a  subtraction  of  the  terms  of  a  ratio. 
{Art.  389.  cor.) 

When  proportions  are  compounded  by  multiplication,  it 
will  often  be  the  case,  that  the  same  factor  will  be  found  in- 
two  analogous  or  two  homologous  terms. 

Thus- if  a:b::c:d) 
And       m : «  : : n : c  ) 


L  J 

am  :  ao  : :  en :  ca 


Here  a  is  in  the  two  first  terms,  and  c  in  the  two  lasiv 
Dividing  by  these,  (Art.  382.)  the  proportion  becomes 

m  :  b  : :  n :  d.  Hence, 

393.    In  compounding  proportions,  equal  factors,  or  divi- 
in  two  analogous-  or  homologous  terms  may  be  rejected. 


200  ALGEBRA. 

(n:b::c:d  12:  4  ::9:3 

Jf  lb:h::cl:l  4  :  8  : :  3  :  6 

th:m::l:n  8:  20::  6: 15 


Then   a:m::c:n  12: 20::  9: 15 

This  rule  may  be  applied  to  the  cases,  to  which  the  terras 
*,' ex  aequo"  and  " ex  aequo  pcrturbate"  refer.  See  arts.  385 
and  387.  One  of  the  methods  may  serve  to  verify  the 
other. 

394.  The  changes  which  may  be  made  in  proportions, 
without  disturbing  the  equality  of  the  ratios,  are  so  nume- 
rous, that  they  would  become  burdensome  to  the  memory, 
if  they  were  not  reducible  to  a  few  general  principles.  They 
are  mostly  produced, 

1.  By  inverting  the  order  of  the  terms,  Art.  380. 

2.  By  mult,  or  dividing  by  the  same  quantity,  Art.  382. 

3.  By  compar.  propor's  which  have  like  terms,  Art.  384,5,6, T, 

4.  By  add.  or  subt.  the  terms  of  equal  ratios,  Art.  388,9. 

5.  By  mult,  or  divid.  one  propor.  by  another,  Art.  390,2,3. 

6.  By  involv.  or  extract,  the  roots  of  the  terms,  Art.  391. 

395.  When  four  quantities  are  proportional,  if  the  first  be 

Jrcater  than  the  second,  the  third  will  be  greater  than  the 
rurth  ;  if  equal,  equal ;  if  less,  less. 

For,  the  ratios  of  the  two  couplets  being  the  same,  if  one 
is  a  ratio  of  equality,  the  other  is  also,  and  therefore  the  an- 
tecedent in  each  is  equal  to  its  consequent ;  (Art.  350.)  if 
one  is  a  ratio  of  greater  inequality,  the  other  is  also,  and 
therefore  the  antecedent  in  each  is  greater  than  its  conse- 
quent ;  and  if  one  is  a  ratio  of  less  inequality,  the  other  is  al- 
so, and  therefore  the  antecedent  in  each  is  less  than  its  con- 
sequent. 

-     •     -     -i 

(a=b,  c—d 
Let  a  :  b  : :  c  :  d ;  then  if  <  cr>  b,  c>  d 


Cor.  1.  If  the  jlrst  be  greater  than  the  third,  the  second 
will  be  greater  than  the  fourth ;  if  equal,  equal :  if  less, 
less.* 

*  Euclid  14.  5. 


PROPORTION,      '  201 

For  by  alternation  a  :  b  :  :  c  :  d  becomes  a  :  c  :  :  b  :  d,  without 
&ny  alteration  of  the  quantities.  Therefore  if  a—b,  c~d,  &c. 
as  before. 

Cor.  2.  If    a  :  m  :  :  c  :  n  )  .,        -f        T         j  B    * 
.    i  7  }  then  if  a=b,  c—d.  ate  * 

And  m  :  b  :\n:d) 

For,  by  equality  of  ratios,  (Art.  385.  2.)  or  compounding 
ratios,  (Arts.  390.  393.) 

a  :  b  :  :  c  :  d.      Therefore,  if  &i=6,  c=d,  &.c.  as  before. 
Cor.  3.  If    a:m  ::n: 


For,  by  compounding  ratios,  (Arts.  390,  393.) 
n  :  b  :  :  c  :  d.     Therefore,  if  «  =±=  b,  c  =  d.  &c. 

CONTINUED    PROPORTION. 

396.  When  quantities  are  in  continued  proportion,  all  the 
ratios  are  equal.  (Art.  372.)     If 

«  :b::btc::c:d::d:e, 

the  ratio  of  a  :b  is  the  same,  as  that  of  ire,  of  c:d,  or  of 
d  :  e.  The  ratio  of  the  first  of  these  quantities  to  the  last^ 
is  equal  to  the  product  of  all  the  intervening  ratios  ;  (Art, 
353.)  that  is,  the  ratio  of  a:  e  is  equal  to 

abed 


But  as  the  intervening  ratios  are  all  equal,  instead  of  mul- 
tiplying them  into  euch  other,  we  may  multiply  any  one  of 
them  into  itself;  observing  to  make  the  number  of  factors 
equal  to  the  number  of  intervening  ratios.  Thus  the  ratio- 
of  a:ct  in  the  example  just  given,  is  equal  to 

* 


\\  hen  several  quantities  are  in  continued  proportion,  the 
aurnberof  couplets,  and  of  course,  the  number  of  ration,  i: 
one  less  than  the  number  of  quantities.  Thus  the  five  pro- 
portional quantities  «?  b,  c,  d,  r,  form  four  eoup]<:tr=  eonlain- 

' 

*  Euclid  £0.  5.  f  Euclid  21.  r,. 

'  •  '.'. 
*JL 


\LGEBRA, 

ing  four  ratios;  and  the  ratio  of  a:c  is  equal  to  the  ratk>  of 
ul:4*,  that  is,  the  ratio  of  the  fourth  power  of  the  first 
quantity,  to  the  fourth  power  of  the  second.  Hrnee, 

397.  If  three  quantities  are  proportional,  the  first  is  to  the 
*/ttV(/,  ay  the  square  of  the  first,  to  the  square  of  the  second;  or 
as  the  square  of  the  second,  to  the  square  of  the  third.  In 
other  words,  the  first  has  to  the  third,  a  dtqilicate  ratio  of  the 
fast  to  the  second. 

If  a:l::b:c,  then  a  :  c  : :  u?  :  I*.     Ami  universally, 

3"98.  If  several  quantities  are  proportional,  the  ratio  of 
the  first  to  the  last  is  equal  to  one  of  the  intervening  ratios 
raised  to  a:  power  whose  index  is  one  less  than  the  number  of 
quantities. 

If  there  are  four  proportionals  a,  &,  c,  d,  then  a :  d: :  a*  :  b3. 
If  there  are  Jive  «,  b,c,(?,e,    a:e::a4  : 64,&,c. 

399.  If  several  quantities  are  proportional,  they  will  be 
proportional  when  the  order  of  the  whole  is  inverted.  This 
has  already  been  proved,  with  respect  to  four  proportional 
quantities.  (Art.  38p.  cor.)  It  may  be  extended  to  any  num- 
ber of  quantities. 

Between  the  numbers,  64,  32,  16,  8,  4, 

The  ratios  are  2,    2,    2,    2,- 

Between  the  same  inverted  4,    8,  16,  32,  64, 

The  ratios  are  ^,    -|,    -|,    £. 

So  if  the  order  of  any  proportional  quantities  be  invert- 
ed, the  ratios  in  one  series  will  be  the  reciprocals  of  those  in 
the  other.  For,  by  the  inversion,  each  antecedent  becomes 
a  consequent,  and  v.  v.  and  the  ratio  of  a  consequent  to  its 
antecedent  is  the  reciprocal  of  the  ratio  of  the  antecedent 
to  the  consequent.  (Art  351.)  That  the  reciprocals  of  equafc 
quantities  are  themselves  equal,  is  evident  from  ax.  4. 


400.  HARMONICAS  os  MUSICAL  PROPORTION  may  be  con- 
sidered as  a  species  of  geometrical  proportion.  It  consists 
in  an  equality  of  geometrical  ratios ;  but  one  or  more  of 
the  terms  is  the  difference  between  two  quantities. 

Three  or  four  quantities  are  said  to  be  in  harmonical  pro- 
portion, when  the  ftrtt  is  to.  the  last,  as  the  difference  ber 


PROPORTION.  203 

the   two  frst,  to  the  difference  between  the  two  last, 

If  the  three  quantities  «,  b,  and  c,  are  in  harmonical  pro- 
portion, then  arc:: a  — b  :b—c. 

If  the  four  quantities  a,  b,  c,  and  <?,  are  in  harmonical  pro- 
portion, then  a:d::a—b:c—d. 

Thus  the  three  numbers  12,  8,  6,  are  in  harmonical  pro- 
portion. 

And  the  four  numbers  20,  16,  12,  10,  are  in  harmonical 
proportion. 

401.  If,  of  four  quantities  in  harmenical  proportion,  any 
three  be  given,  the  other  may  be  found.  For,  from  the 
proportion 

a:d::a  —  b:c—d, 

by  taking  the  product  of  the  extremes  and  the  means,  we 
have  ac—ad=^ad—bd. 

And  this  equation  may  be  reduced,  so  as  to  give  the  valire 
of  either  of  the  four  letters. 

Thus  by  transposing  —  ad,  and  dividing  by  «5 

2nd -Id 


402.  A  list  of  the  articles  in  this  section  which  contain  the 
propositions  in  the  5th  book  of  Euclid.* 


Prop.   I. 

Art.  363. 

XIII. 

384,  cor. 

II. 

388. 

XIV. 

395,  cor.  1, 

HI. 

382. 

-XV. 

360. 

IV. 

382,  cor.  1. 

XVI. 

380. 

V. 

362. 

-XVII. 

389,  cor. 

VI. 

362. 

XVIII. 

389,  2. 

\VII. 

349,  cor.  1. 

XIX. 

389,  4. 

VIII. 

357,  cor.  358, 

cor.    XX. 

395,  cor.  2, 

XIX. 

349,  cor.  2. 

XXI. 

395,  cor.  3. 

X. 

357,  cor.  358, 

cor.     XXII. 

386. 

XI. 

3S4. 

XXIII. 

387. 

-XII. 

363. 

XXIV. 

388,  cor.  2. 

*  See  Note  I. 


VARIATION  OR  GENERAL  PROPORTION 


A  dO4?  T^JE  quantities  which  constitute  the  u nr,s  oi 
-  a  proportion  are,  frequently,  so  related  to 
.each  other,  that,  if  one  of  them  be  either  increased  or  dimin- 
ished, another  .depending  on  it  will  also  be  increased  or  di- 
minished, iu  such  a  manner,  that  the  proportion  will  still  he 
preserved.  If  the  value  of  50  yards  of  cloth  is  100  dollars, 
and  the  quantity  be  reduced  to  40  yards ;  the  value  will,  of 
.course,  be  reduced  to  80  dollars :  if  the  quantity  be  reduced 
•io  30  yards,  the  value  will  be  reduced  to  GO  dollars,  &tc. 

yd.  yd.     dol.  dol. 
That  is     SO: 40:;  100: 80     , 
50:  30::  100: 60 
SO :  20  : :  100  : 40,  &c. 

As  the  consequent  of  the  first  couplet  is  varied,  the  .con- 
sequent of  the  second  is  varied,  in  such  a  manner,  that  the 
proportion  is  constantly  preserved. 

If  the  two  antecedents  are  A  and  B ;  and  if  a  represents 
3.  quantity  of  the  same  kind  with  Jl,  but  either  greater  or  less ; 
and  by  a  quantity  of  the  same  kind  with  B,  but  as  many  times 
greater  or  less,  as  a  is  greater  or  less  than  A ;  then 

A:a::B:b', 

that  is,  if  A  by  varying  becomes  a,  then  B  becomes  l\  This 
is  expressed  more  concisely,  by  saying  that  Jl  varies  as  //',  or 
A  is  as  B.  Thus  the  wages  of  a  labouring  man  vary  as  the 
time  of  his  service.  We  say  that  the  interest  of  mon^y 
which  is  loaned  for  a  given  time,  is  proportioned  to  the  prin- 
cipal. But  a  proportion  contains  four  terms.  Here  are  only 
two,  the  interest  and  the  principal.  This  then  is  an  abrirfg- 
rd  statement,  in  which  two  terms  are  mentioned  instead  of 
four.  The  proportion  in  form  would  be  j 

*  Newton's  Princlp.  Book  i.  Sec.  1.  Lemma  10,  scliol.  Emerson 
on  Proportion,  Wood's  Algebra,  Ludlam's  Math.  Saunderson's  Alge- 
jbra,  Art.  £30, 


VARIATION.  «r>3 

As  any  given  principal,  is  to  any  other  principal  ; 

So  is  the  interest  of  the  former,  to  the  interest  of  the  latter. 

P.     P.      In.  In. 

For  instance,     1  00  :  300  :  :  G  :  1  8. 

£04.  In  many  mathematical  and  philosophical  investiga- 
tions, we  have  occasion  to  determine  the  general  relations  of 
certain  classes  of  quantities  to  each  other,  without  limiting 
the  inquiry  to  any  particular  values  of  those  quantities.  In 
such  cases,  it  is  frequently  sufficient  to  mention  only  two  of 
the  terms  of  a  proportion.  It  must  be  kept  in  mind,  howev- 
er, that  four  are  always  implied.  When  it  is  said,  for  instance, 
that  the  weight  of  water  is  proportioned  to  its  bulk,  we  arc 
lo  understand. 

That  one  gallon,  is  to  any  number  of  gallons; 
As  the  weight  of  one  gallon,  is  to  the  weight  of  the  given 
number  of  crallons. 

O 

405.  The  character  <x  is  used  to  express  the  proportion 
of  variable  quantities. 

Thus  vIoaB  signifies  that  Jl  varies  as  B,  that  is,  that 


:a:;'.. 

The  expression  &&B  may  be  called  a  general  proportion. 

406.  One  quantity  is  said  to  vary  directly  as  another,  when 
the  one  increases  as  the  other  increases,  or  is  diminished  as 
the  other  is  diminished,  so  that 

«#<*#,  that  is  ./2:a  ;:#:&. 

The  interest  on  a  loan  is  increased  or  diminished,  in  pro- 
portion to  the  principal.  If  the  principal  is  doubled,  the  in- 
terest is  doubled;  if  the  principal  is  trebled,  the  interest  is 
trebled,  &c. 

407.  One  quantity  is  said  to  vary  inversely  or  reciprocally 
as  another,  when  the  one  is  proportioned  to  the  reciprocal  of 
the  other;  that  is,  when  the  one  is  diminished,  as  the  other 
is  increased,  so  that 

'  •ft''a::  7?  :  T>  or«#:or  ::£:  J5. 

In  this  case,  if  A  is  greater  than  a,  B  is  less  than  Z».  (Art. 
1305.)  The  time  required  for  a  man  to  raise  a  given  sum,  by 
his  labour,  is  inversely  as  his  wages.  The  higher  his  wuges, 
the  less  the  time. 

-'.08.  One  quantity  is  said  to  vary  as  two  plJirra  jninthf,  when 


205  ALGEBRA. 

the  one  is  increased  or  diminished,  as  the  ptoduct  of  the 
other  two,  so  that  (' 

A  «  JBC',  that  is  A  :  a  ;  :  B  C  :  be. 

The  interest  of  money  varies  as  the  product  of  the  prin- 
cipal and  time.  If  the  time  IK-  doubled,  and  the  principal 
doubled,  the  interest  will  be  four  times  as  great. 

409.  One  quantity  is  said  to  vary  directly  as  a  second,  and 
inversely  as  a  third,  when  the  first  is  always  proportioned  to 
the  second  divided  by  the  third,  so  that 

B  B    b 

~ni  that  is  A  :  a  :  :      :  —  * 


410.  To  understand  the  methods  by  which  the  statements 
of  the  relations  of  variable  quantities  are  changed  from  one 
form  to  another,  little  more  is  necessary,  than  to  make  an 
application  of  the  principles  of  common  proportion;  bear- 
ing constantly  in  mind,  that  a  general  proportion  is  only  an 
abridged  expression,  in  which  two  terms  are  mentioned  in- 
stead of  four.     When  the  deficient  terms  are  supplied,  the 
reason  of  the  several  operations  will  commonly  he  apparent. 

411.  It  is  evident,  in   the  first  place,  that  the  order  of  the. 
terms  in  a  general  proportion  may  be  inverted,  (Art.  369.) 


If        A:a::B:b,  that  is,  ife 
Then  B  :  b  :  :  A  :  a,  that  is,     B  <*J1. 


412.  If  one  or  both  the  terms  in«,  general  proportion,  be 
multiplied  or  divided  by  a  constant  quantity,  the  proportion 
will  be  preserved. 

For  multiplying  or  dividing  one  or  both  of  the  terms  is 
the  same,  as  multiplying  or  dividing  analagous  terms  in  the  . 
proportion  expressed  at  length.  (Art.  382.  and  cor.  1.) 

If  A:a::B:b,       that  is,  if       J1&B, 

Then  fl!i#:m«  ::!?:&,       that  is,        mA<*-TB, 

And    mJl:ma::inB:mb,  that  is,        mJl^mB.  &c. 


41  3.  If  both  the  terms  be  multiplied  or  divided  even  by 
a  variable  quantity,  the  proportion  will  be  preserved.  For 
this  is  equivalent  to  multiplying  the  two  antecedents  by  one 
quantity,  and  the  two  consequents  by  another.  (Art.  382.) 

If  ./2:   a   ::  B    :  b,  that  is,  if      .tfoc-S; 

Then  MA  :ma  ::  MB  :  mb,  that  is,      MrfouMB,  &c. 
Cor.  1.  If  ope  quantity  varies  as  another,  the  quotient  of 

^«  '•{  1&   ><^vv.  •  .  '  -i.  f/-<*H  i 


VARIATION.  3?07 

the  one  divided  by  the  other  is  constant.  In  other  words,  if 
me  numerator  of  a  fraction  varies  as  the  derfominator,  the 
value  remains  the  same. 

If  A-.  <r::B:  b,  that  is,  if  A&B, 
A     a    B     b 

Then  s:T':B:J'':1''1'     (Art-  128-) 

Here  the  third  and  fourth  terms  are  equal,  because  e"ach 
is  equal  to  1.  Of  course  the  two  first  terms  are  equal  ;  (Art. 
395.)  so  that,  if  A  be  increased  or  diminished  as  many  times' 
as  Br  the  quotient  will  be  invariably  the  same. 

Cor.  2.  If  the  product  of  two  quantities  is  constant,  one 
vanes  reciprocally  as  the  other. 

™    „„     7  JIB.  ab       1      I  11 

If  Jin  :  ab  :  :  1  :  1  ,  then  -jr-  :  "T  :  :  "n  :  T>  or  **  :  a  '•  '•  ~n  '  ~T 

Cor.  3.  Any  factor  in  one  term  of  a  general  proportion, 
may  be  transferred,  so  as  to  become  a  divisor  in  the  other  f 

and  v,  v. 

a 

If  ^ocBC,then  dividing  by  if,  ^ccC.  (Art.  118.) 
If  A  <x  pjj,  then  mult,  by  C,£C=p-  (Art.  159.) 


414.  If  two  quantities  vary  respectively  as  a  third,  then 
of  the  two  varies  as  the  other.  (Art.  384.) 


If  A\a::B'.l\a  ..  .,  <AXB 
&  i  r<  T?  i  f  that  is,  it  <  r<  7, 
And  C:c::B:b<)  (Coc/?; 

Then     ^  :o::  C:c,  that  is, 


415.  If  two  quantities  vary  respectively  as  a  third,  their 
nv.m  er  difference  will  vary  in  the  same  manner.  (Art.  388.) 

If  A:a::B:blt,  '  .  ., 
A  r  r>  »  7  f  that  is,  it 
And  C:c::13:b 


Then  .^+  C:«  +  e::B:&,  that  is, 
And    Jl-C-.a-c-.B-.b,  that  is,  A- 

Cor.  The  addition  here  may  be  extended  to  any  number 
uf  quantities  all  varying  alike.  (Art.  3S8.  cor.  1.) 
If  *?cc5,  and  Ccc#,  and  I>KB,  and  ECC#,  then 


.*+\t  ^.K-r  wj--*4r-|-.«y  »>.«. 

416.  The  terms  of  one  general  proportion  may  be  multi- 
plied or  divided  bv  the  corresponding  terms  of  another 
(Ait.  390.) 

•   9~ 

• 


ALGEBRA. 


If 

And 


Then    AC:ac::BD:bd  that  is,  AC*BD. 

Cor.  If  two   quantities  vary  respectively  as  a  third,  the 
product  of  the  two  will  vary  as  the  square  of  the  other. 


417.  If  any  quantity  vary  as  another,  any  power  or  root 
of  the  former  will  vary,  as  a  like  power  or  root  of  the  latter. 
(Art.  391.) 


If        A:a::B:b,          that  is,  if 

Then  An  :  an  :  :  Bn  :  b*      that  is,         An  oc  Bn  , 

And    An:a"::B":ln,    that  is,         .#"«£". 

418.  In  compounding  general  proportions,  equal  factor* 
Or  divisors,  in  the  two  terms,  maybe  rejected.  .(Art  393.) 


If     A:a::B:b) 

And  B  :  b  ::  C  :  c  >  that  is,  if  <B  «  C 
AndC:c::D:d 


Then      tf:a::D:d,      that  is, 

Cor.  If  one  quantity  varies  as  a  second,  the  second,  as  a 
third,  the  third,  as  a  fourth,  Sic.  then  the  first  varies  as  the 
last. 

If  AxBx  CccD,  then  Ax  D. 

If  «/?<x.BoC7i,then.#ce-77;  that  is,  if  the  first  varies  direct!  >/ 

as  the  second,  and  the  second  varies  reciprocally  as  the  third  ; 
the  first  varies  reciprocally  as  the  third. 

1^  419.  If  any  quantity  vary  as  the  product  of  two  others. 
and  if  one  of  the  latter  be  considered  constant,  the  first  will 
vary  as  the  other. 

If  JFcc  LB,  and  if  B  be  constant,  then  W&  L. 

Here  it  must  be  observed,  that  there  are  two  conditions: 
First,  that  W  varies  as  the  product  of  the  two  other  quan- 

tities; 
Secondly,  that  one  of  these  quant  Jtio^  B  i.s  constan'. 


VARIATION.  209 

Then,  by  the  conditions,  W:w::  LB  :lBjB  being  the  same 

in  both  terms. 

Divid.  by  the  constant  quantity  B,  W\w::L:  /,  that  is  W&.L. 
And  if  L  be  considered  constant,  W<x.  B. 

Thus  the  weight  of  a  board,  of  uniform  thickness  and  den- 
sity, varies  as  its  length  and  breadth.  If  the  length  is  given, 
the  weight  varies  as  the  breadth.  And  if  the  breadth,  is  giv- 
en, the  weight  varies  as  the  length. 

Cor.  The  same  principle  may  be  extended  to  any  number 
of  Quantities.  The  weight  of  a  stick  of  timber,  of  given 
density,  depends  on  the  length,  breadth,  and  thickness.  If 
-the  length  is  given,  the  weight  varies  as  the  breadth  and 
thickness.  If  the  length  and  breadth  are  given,  the  weight 
.varies  as  the  thickness,  &c. 

If  WaLBT; 

Then  making  L  constant,  Wrx.B.T', 

And  making  L  and  B  constant,  W<*  &J^ 

420.  On  the  other  hand,  if  one  quantity  depends  on  two 
others;  so  that  when  the  second  is  given,  the  first  varies  'as 
the  third,  and  when  the  third  is  given,  the  first  varies  as  the 
second;  then  the  first  varies  as  the  product  of  the  other 
two. 

If  the  weight  of  a  board  varies  as  the  length,  when  the 
ibreadth  is  given,  and  as  the  breadth  when  the  length  is  giv- 
*n ;  then  it  the  length  and  breadth  both  vary,  the  weight  va- 
ries as  their  product 

If      1V<xL,  when  B  is  constant,  >  ,        -irr    pr 
And  WxB,  when  L  is  constant!  \  then  W«BL- 

In  demonstrating  this,  we  have  to  consider,  two  variable 
values  of  W ;  one,  when  L  only  varies,  and  the  other,  whcu 
L  and  B  both  vary. 

Let  «>'=the  first  of  these  variable  values, 
Andw=the  other; 

So  that  W  will  be  changed  to  w',  by  the  varying  of      L ; 
And  w  will  be  farther  changed  to  «?,  by  the  varying  of  B. 

Then,  by  the  supposition,  W: iv'  ::L:l,  when  B  is  constant. 

And  w' :  w  : :  B  :  b,  when  B  varies. 

Mult,  correspond,  terms,     Ww' :  ww  : :  BL :  bl.  (Art.  390.) 
Divid.  by  w'  (Art.  382.)     W :  w  ::BL:  bl,  i.  e. .  /F«  BL 

Cc 


ALGEBRA. 

The  proof  may  be  extended  to  any  number  of  quantities. 
The  weight  of  a  piece  of  timber,  depends  on  its  length, 
breadth,  thickness  and  density.  If  any  three  of  these  are 
given,  the  weight  varies  as  the  other. 

This  case  must  not  be  confounded  with  that  in  art.  416, 
cor.  In  that,  B  is  supposed  to  vary  as  Jl  and  as  C\  at  the 
same  time.  In  this,  B  varies  as ./?,  only  when  C  is  constant, 
and  as  C,  only  when  A  is  constant.  It  can  n»t  therefore  va- 
yy  as  A  and  as  C  separately,  at  the  same  time. 


-A 


421.  Many  writers,  in  expressing  a  general  proportion,  d» 
not  use  the  term  vary,  or  the  character  which  has  here  been 
put  for  it.  Instead  of  JlctB,  they  say  simply  that  A  is  asB. 
See  Enfield's  Philosophy.  It  may  be  proper  to  observe  al- 
so, that  the  word  given  is  frequently  used  to  distinguish  con- 
stant quantities,  from  those  which  are  variable;  as  well  as  to 
/distinguish  known  quantities,  from  those  which  are  unknown. 
(Art,  17.) 


fl£. 
4v    tivfa-^  '^1 


•<rt 


,/Aj  '/Hitfh' 

.  //W~: 

^-VM  V 


,.       i 


.    ff  -  >j 

/i  <l(  ^ 

l».tniyh 


SECTION    XIV. 


ARITHMETICAL  AND  GEOMETRICAL  PROGRES- 
SION. 


A  /199  OUANTtTlES  which  decrease  by  a  common 
*&  difference,  as  the  numbers  10,  8,  6,  4,  2,  are 
in  continued  arithmetical  proportion.  (x\rt.  372.)  Such  a 
series  is  also  called  a  progression,  which  is  only  another  name 
for  continued  proportion. 

It  is  evident,  that  the  proportion  will  not  be  destroyed,  if 
the  order  of  the  quantities  be  inverted.  Thus  the  numbers 
2,  4,  6,  8,  10,  are  in  continued  arithmetical  proportion. 

Quantities,  then,  are  in  arithmetical  progression,  when  they 
increase  or  decrease  by  a,  common  difference* 

When  they  increase,  they  form  what  is  called  an  ascending 
series,  as  3,  5,  7,  9,  11,  &c, 

When  they  decrease,  they  form  a  descending  series,  as  11, 
9,  7,  5,  &c. 

The  natural  numbers  1,  2,  3,  4,  5,  6,  &tc.  are  in  arithmetic- 
al progression  ascending. 

423.  From  the  definition  it  is  evident  that,  in  an  ascend- 
ing series,  each  succeeding:  term  is  found,  by  adding  the  com- 
mon difference  to  the  preceding  term. 

If  the  first  term  is  3,  and  the  common  difference  2; 
Then  3+2=5  is  the  second  term,  7  +  2=9  the  fourth. 

5+2=7    the  third,  9+2=11  the  fifth,  &tc, 

And  the  series  is  3,  5,  7,  9,  11,  13,  &c. 

If  the  first  term  is  a,  and  the  common  difference  d; 

Then  a+d  is  the  second  term,  «  +  2d+fl?=a+3^  the  fourth, 
the  3d,  a+3d+d=a+4ihhe  5th,SiC. 


123  4  5 

And  the  series  is  a,  a  +  d,  a+2d,  a  +  3d,  a  +  4^?,  &c. 

If  the  first  term  and  the  common  difference  are  the  tame. 


21J3T  ALGEBRA. 

the  series  becomes  more  simple.     Thus  if  a  is  the  first  terro, 
and  the  common  difference,  and  n  the  number  of  terms. 

Then  a  +  a=2a  is  the  second  terra 
2«-f  «=3a  the  third,  &c. 

And  the  series  is  a,  2a,  3a,  4a  .....  na* 

424.  In  a  descending  series,  each  succeeding  term  is  found, 
by  subtracting  the  common  difference  from  the  preceding 
term. 

If  a  is  the  first  term,  and  d  the  common  difference,  the 

123  4  S 

series  is  a,  a—d,  a—2d,  a—  3d,  a—  4J,  &c. 

In  this  manner,  we  may  obtain  any  term,  by  continued  ad- 
dition or  subtraction.  But  in  a  long  series,  this  process  would 
become  tedious.  There  is  a  method  much  more  expedi- 
tious. By  attending  to  the  series 

125  4  S 

a,  a+d,  a+2d.  a+3d,  a+4J,  &sc. 

it  will  be  seen,  that  the  number  of  times  d  is  added  to  a  A- 
one  less  than,  the  number  of  the  term. 

The  second  term  is  a  +  d,  i.  e.  a  added  to  once  d; 
The  third  is  a+2d,       a  added  to  twice  d; 

1?he  fourth  is  a  +  3d,       a  added  to  thrice  d,  &GV 

So,  if  the  series  be  continuedr 

The  60th  term  will  be  a+49d 

The  100th  term 


In  the  greatest  term,  the  number  of  times  dis  added  to*  c, 
is  one  less  than  the  number  of  all  the  terms.     If  then 
n=  the  least  term,  z=the  greatest,  n=  the  number  of  terms, 
vie  shall  have,  in  all  cases,  z  —a  +(»  —  1)  X  d;  that  is, 

425.  In  an  arithmetical  progression,  the  greatest  term  is 
equal  to  the  least,  -J-  the  product  of  the  common  difference  into 
the  number  of  terms  lets  one. 

Any  other  term  may  be  found  in  the  same  way.  For 
the  series  may  be  made  to  stop  at  any  term,  and  that  maj 
be  considered,  for  the  time,  as  the  last. 

Thus  tbe  with  term=o+(m—  1)  X  d. 

If  the  first  term  and  the  common  difference  are  the  same, 

z—a+(n  —  l)a=a+na—a,  that  is,  z=na. 
Li  an  ascending  series,  the  first  term  's,  evidently,  the  least, 


ARITHMETICAL  PROGRESSION.         213 

uml  the  last,  the  greatest.      But  in  a  descending  series,  the 
first  term  is  the  greatest,  and  the  last,  the  least. 

426.  The  equation  z—a-\-(n—l}d  not  only  shows  the  val- 
ue of  the  greatest  term,  but,  by  a  few  simple  reductions,  will 
enable  us  to  find  other  parts  of  the  series.  It  contains  four 
different  quantities, 

a,  the  least  term,         n,  the  number  of  terms,  and 
z,  the  greatest  term,    d,  the  common  difference. 

If  any  three  of  these  be  given,  the  other  may  be  found. 

1.  By  the  equation  already  found, 

#  =  a  +  (w  —  l)d=the  greatest  term. 

2.  Transposing  (n  -l)d,  (Art.  173,)' 

z  —  (n  —  1  }d  =  a  —  the  least  term. 

3.  Transposing  a  in  the  1st  and  dividing  by  n—  I, 

z  —  a 


—d=the  common 


n-l 

4.  Transp.  a  in  the  1st,  dividing  by  d,  and  transp.  —I, 

z — a 

— -7—  -f  1  =n=fAe  number  of  terms. 

Prob.  1.  If  the  first  term  of  an  increasing  progression  is 
7,  the  common  difference  3,  and  the  number  of  terms  9, 
what  is  the  last  term? 

Ans.  z—a+(n— l)J=7  +  (9  —  I)x3=31. 
And  the  series  Is  7,  10,  13,  16,  19,  22,  25,  28,  31. 

Prob.  2.  If  the  last  term  of  an  increasing  progression  IK 
<50,  the  number  of  terms  12,  and  the  common  difference  5, 
what  is  the  first  term  ? 

Ans.  a=;z—  (n  —  1)J=60  —  (12— I)x5=5. 

•^ 

427.  There  is  one  other  inquiry  to  be  made  concerning  a 
series  in  arithmetical  progression.  It  is  often  necessary  to 
find  the  sum  of  all  the  terms.  This  is  called  the  summation  of 
the  series.  The  most  obvious  mode  of  obtaining  the  amount 
of  the  terms,  is  to  add  them  together.  But  the  nature  of 
progression  will  furnish  us  with  a  method  more  expeditious.  '* 

It  is  manifest  that  the  sum  of  the  term*  will  be  the  samer 


214  ALGEBRA. 

in  whatever  order  they  are  written.  The  sum  of  the  ascend-- 
ing  series  3,  5,  7,  9,  11,  is  the  same,  as  that  of  the  descend- 
ing series  11,  9,  7,  5,  3.  The  sum  of  both  the  series  is,  there- 
fore, tirice  as  great,  as  the  sum  of  the  terms  in  one  of  them. 
There  is  an  easy  method  of  finding  this  double  sum,  and,  of 
course,  the  sum  itself  which  is  the  object  of  inquiry.  Let  a 
given  series  be  written,  both  in  the  direct,  and  in  the  invert- 
ed order,  and  then  add  the  corresponding  terms  together. 

Take,  for  instance,  the  series  3,    5,    7,    9,  11, 

And  the  same  inverted  11,    9,    7,    5,    3. 

The  sums  of  the  terms  will  be         14,  14,  14,  14,  14. 

Take  also  the  series     a,  a  +  rf,     a+2d, 

And  the  same  inver.     a +4(7,  a  +  3d,  a  +  2d, 


The  sums  will  be        2a  +  4rf,2a  +  4d,2a  +  4d,2a  +  4e/,2a  +  4rf. 

Here  we  discover  the  important  property,  that, 

428.  In  an  arithmetical  pi'ogression,  the  sum  of  the  extremes 
is  equal  to  the  sum  of  any  other  two  terms  equally  distant  from 
the  extremes. 

In  the  series  of  numbers  above,  the  sum  of  the  first  and 
the  last  term,  of  the  first  but  one  and  the  last  but  one,  &c.  is 
14.  And  in  the  other  series,  the  sum  of  each  pair  of  cor- 
responding terms  is  2a+4df. 

To  find  the  sum  of  all  the  terms  in  the  double  series,  we 
have  only  to  observe,  that  it  is  equal  to  the  sum  of  the  ex- 
tremes repeated  as  many  times  as  there  are  terms. 
The  sum  of  14,  14,  14,  14,  14=14x5. 

And  the  sum  of  the  terms  in  the  other  double  series  i* 


But  this  is  twice  the  sum  of  the  terms  in  the  single  serie0. 
If  then  we  put 

a=the  least  term,  n=the  number  of  terms, 

r=the  greatest,  s=the  sum  of  the  terms. 

we  shall  have  this  equation, 

a+z 
s——£-xn.     That  is, 

429.  In  an  arithmetical  progression,  the  sum  of  all  the. 
terms  is  equal  to  half  the  sum  of  the  extremes  multiplied 
the  number  of  terms. 


ARITHMETICAL  PROGRESSION.          215 


Pi'ob.  What  is  the  sum  of  the  natural  series  of  numbers 
1,2,3,  1,5,  &c.upto  1000? 

+z  1  +  1000 

= 


\ 

' 


Ans.  »=—  5-x  n  =  -  „  --  x  1000=500500. 


430.  In  the  series  of  odd  numbers  1,  3,  5,  7,  9,  &,c.  con- 
tinued to  any  given  extent,  the  last  term  is  always  one  less 
than  twice  the  number  of  terms. 

For  z—a-\-(n  —  l)d.  (Art.  425.)     But  in  the  proposed  se- 
ries a  =  l,  and  d=2. 

The  equation,  then,  becomes  z  =  l  +  (n  —  I)x2=2n  —  1. 

431.  In  the  series  of  odd  numbers  1,  3,  5,  7,  9,  &c.  the 
sum  of  the  terms  is  always  equal  to  the  square  of  the  number  of 
terms. 

For  s=-|(fl+z)n.  (Art,  429.) 
But  here  «  =  1,  and  by  the  last  article,  z=2n—  1. 
The  equation,  then  becomes,  s=4-(l  +  2,n  —  l)n=n9. 

Thus  1+3=4   } 

1+3+5=9    >  the  square  of  the  number  of  terms. 
1+3+5+7  =  16  i 

Or  thus, 

Series  of  numbers,  1,  3,  5,  7,  9,  11,  13,  15,  &.c. 
Number  of  terms,  1,  2,  3,  4,  5,  6,  7,  8,  &c. 
Sum  of  the  terms,  1,  4,  9,  16,  25,  36,  49,  64.  &c. 

432.  If  there  be  two  ranks  of  quantities  in  arithmetical 
progression,  the  sums  or  differences  will  also  be  in  arithmetical 
progression. 

For,  by  the  addition  or  subtraction  of  the  corresponding 
terms,  the  ratios  are  added  or  subtracted.  (Art.  345.)  And 
by  the  nature  of  progression,  all  the  ratios  in  thfe  series  are 
equal.  Therefore  equal  ratios  being  added  to,  or  subtracted 
from,  equal  ratios,  the  new  ratios  thence  arising  will  also  be 
equal. 

To  and  from     3,    6,    9,12,15,18,20  f3 

Add  and  sub.    2,    4,    6,    8,  10,  12,  14  |  |  2 

---  >  whose  ratio  is  ^  — 
Sums  5,  10,  15,  20,  25,  30,  35  15 

DifF.  1,    2,    3,    4,    5,    6,    7J  [l 

,433.  If  all  the  terms  of  an  arithmetical  progression  be 


216  ALGEBRA. 

multiplied  or  divided  by  the  same  quantity,  the  products  or 
quotients  will  be  in  arithmetical  progression. 

For,  by  the  multiplication  or  division  of  the  teftns,  the 
ratios  are  multiplied  or  divided;  (Art.  344.)  that  is,  equal 
quantities  are  multiplied  or  divided  by  the  given  quantity. 
They  will  therefore  remain  equal. 

If  the  series  3,  5,  7,  9, 11, Sic.  be  mult,  by  4; 

The  prods,  will  be  12-,20,28,36,44,kc.aji4ifthisbedivid.by2; 

The  quots.  will  be    6,10,14,l8,22,&c.  jf 


GEOMETRICAL   PROGRESSION. 

434.  As  arithmetical  proportion  continued  is  arithmetical 
progression,  so  geometrical  proportion  continued  is  geomet- 
rical progression. 

The  numbers  64,  32,  1C,  8,  4,  are  in  continued  geomet- 
rical proportion.  (Art.  372.) 

In  this  series,  if  each  preceding  term  be  divided  by  the 
common  ratio,  the  quotient  will  be  the  following  term. 
V=32,  and  y  =16,and  y  =8,  and  |=4. 

If  the  order  of  the  series  be  inverted,  the  proportion  will 
still  be  preserved;  (Art.  399.)  and  the  common  divisor  will 
become  a  multiplier.  In  the  series 

4,8,16,32,64,  &c.  4 x  2=8,  and  8  X  2=16,  and  16  x  2=*32,&c. 

435.  Quantities,  then,  are  in  geometrical  progression,  when 
they  increase  by  a  common  multiplier,  or  decrease  by  a  com- 
mon divisor. 

The  common  multiplier  or  divisor  is  called  the  ratio.  In 
a  descending  series,  it  is,  as  in  common  proportion,  a  direct 
ratio.  But  in  an  ascending  series,  it  is  a  reciprocal  ratio. 

In  the  scries  4,8,1 6,32,&c.4  =  Y  =**-=2,the  common  ratio. 

But  if  the  direct  ratios  are  equal,  the  reciprocal  ratios  are 
also  equal.  (Art.  399.)  So  that  quantities  in  geometrical  pro- 
gression, whether  ascending  or  descending,  may  be  consid- 
ered proportionals. 

To  investigate  the  properties  of  geometrical  progression, 
we  may  take  nearly  the  same  course,  as  in  arithmetical  pro- 
gression, observing  to  substitute  continual  multiplication  and 
division,  instead  of  addition  and  subtraction.  It  is  evident,  in 
the  first  place,  that, 


GEOMETRICAL  PROGRESSION.  217 

436.  In  an  ascending  geometrical  series,  each  succeeding 
term  is  found  by  multiplying  the  ratio  into  the  preceding  term. 

If  the  first  term  is  a,  and  the  ratio  r, 

Then  axr=ar,  the  second  term,  ar2  xr=er3,  the  fourth, 
ar x  r=ars,  the  third,  ar3  xr=ar4,  the  fifth,  &c. 

And  the  series  is  a,  ar,  «r3,  ar3,  ar4,  ar*,  &c. 

437.  If  the  first  term  and  the  ratio  are  the  same,  the  pro- 
gression is  simply  a  series  of  powers. 

If  the  first  term  and  the  ratio  are  each  equal  to  r, 

Then  rxr=r2,  the  second  term,      r3  Xr=r4,  the  fourth, 

rs  xr=r3,  the  third  ,  r*xr=r5,  the  fifth. 

And  the  series  is  r,  r3,r3,  r4,  r5,  r6,  &c. 

438.  In  a  descending  series,  each  succeeding  term  is  found 
by  dividing  the  preceding  term  by  the  ratio. 

If  the  first  term  is  ar6,  and  the  ratio  r, 

The  series  is  ar6,  ar5,  ar4,  or3,  ar8,  ar,  a,  &e. 

If  the  first  term  is  a  and  the  ratio  r, 

a      a      a 
The  series  is  «,  —  >  ^>  ^j,&c,or(Art.207.)a,ar~l,or*-s,&;e. 

If  the  first  term  is  1 ,  and  the  ratio  2, 
The  series  is  1,  |,  £,  j,  TV,  •&,  ?\:  &e. 

123  4  5  6 

By  attending  to  the  series  a,  ar,  ars,  ar3,  ar4,  ar*,  &c.  it 
will  be  seen  that,  in  each  term,  the  exponent  of  the  power  of 
the  ratio  is  one  less,  than  the  number  of  the  term. 

vlf  then  a=the  least  term,         r=the  ratio 

^=the  greatest,  7z=the  number  of  terms; 

we  have  the  equation  z=ar"~1,  that  is, 

439.  In  geometrical  proguession,  the  greatest  term  is  equal 
to  the  product  of  the  least,  into  that  power  of  the  ratio  ivhose 
index  is  one  less  than  the  number  of  terms. 

When  the  least  term  and  the  ratio  are  the  same,  the  equa- 
tion becomes  z=rrn~l~rn .  See  art.  437. 

440.  Of  the  four  quantities  a,  z,  r,  and  n,  any  three  being- 
given,  the  other  may  be  found.* 

*See  Note  K. 
Dd 


i*  ALGEB11A. 

1.  By  the  last  article, 

z=ar"~l=ihe  greatest  term. 

2.  Dividing  by  rn~l, 

z 
—  -rj-=^=the  least  term. 

3.  Dwid.  the  1st  by  a,  and  extracting  the  root,  (Art.  297.) 

fz  \«li 

W      =r=the 


441.  Tlie  next  thin    to  be  attended  to  is  the  rule  for  find- 


iiitt the  sum  of  all  the  terms. 

If  any  term,  in- an  increasing  geometrical  series,  be  multi- 
plied by  the  ratio,  the  product  will  be  the  succeeding  term. 
(Art.  436.)  Of  course,  if  each  of  the  terms  be  multiplied 
by  the  ratio,  a  new  series  will  be  produced,  in  which  all  the 
terms  except  the  last  will  be  the  same,  as  all  except  the  first 
in,  the  other  series.  To  make  this  plain,  let  the  new  series 
be  written  under  the  other,  in  such  a  manner,  that  each 
term-  shall  be  removed  one  step  to  the  right  of  that  from 
which  it  is  produced  in  the  line  above. 

Take,  for  instance,  the  series  2,  4,  8,  16,  32 

Mult,  each  term  by  the  ratio,  we  have  4,  8, 16,  32, 64. 

Here  it  will  be  seen,  at  once,  that  the  four  last  terms  in 
the  upper  line  are  the  same,  as  the  four  first  in  the  lower 
line.  The  only  terms  which  are  not  in  both,  are  the  first  of 
the  one  series,  and  ilielast  of  the  other.  So  that  when  we 
subtract  the  one  series  from]  the  other,  all  the  terms  except 
these  two  will  disappear,  by  balancing  each  other. 


If  the  given  series  is         «,  ar,  ar2,  or3,  —  ar*~l. 
Then  mult,  by  r,  we  have,     ar,  ar2,  ar8, ....  arn~1,  arn . 

Now  let  s  =the  sum  of  the  terms, 

Then  s=a+ar+ar~+ar3, —  +  arn~l, 

And  mult,  by  r,  rs=       ar+ar24-ar3,  ....+«'"""* +  crrf! . 

Subt'g  the  first  equation  from  the  second,     rs—  s=arn  —a 

tlTn  —tt 

And  dividing  by  (r-1),  (Art.  121.)  s=~^~ "f"  * 


GEOMETRICAL  PROGRESSION.  219 

In  this  equation,  «rn  is  the  last  term  in  the  new  series,  and 
is  therefore  the  product  of  the  ratio  into  the  last  term  in  the 
given  series.^  Ii«wwp, 

442.  Thesum  of  a  series  in  geometrical  progression  is 
found,  by  multiplying  the  greatest  term  into  -the  ratio,  sub- 
tracting the  least  term,  and  dividing  the  remainder  by  the 
ratio  less  one. 

%Prob.  If  in  a  series  of  numbers  in  geometrical  progres- 
sion, the  first  term  is  6,  the  last  term  1458;  and  the  ratio  3, 
what  is  the  sum  of  all  the  terms  ? 

rz— a     3x1458—6 

Ans.  s— —    T-= 5— ^ =2184. 

r—  I  3—1 


443.  Quantities  in  geometrical'  progression  are  proportional 
to  their  differences. 

Let  the  series  be  a,  or,  or2,  or3,  or4,  Stc. 
By  the  nature  of  geometrical  progression, 

a  :  or : :  ar :  ar2  : :  ar2  :  ar3  : :  or3  :  or4,  &zc. 

In  each  couplet  let  the  antecedent  be  subtracted  from  the 
consequent  according  to  art.  389.  6. 

Then  a:ar::or— a:  or2  —ar  ::  ar2  —ar:ar*—ar2,  &c. 

That  is,  the  first. term  is  to  the  second,  as  the  difference  be- 
tween the  first  and  second,  to  the  difference  between  the 
second  and  third ;  and  as  the  difference  between  the  second 
and  third,  to  the  difference  between  the  third  and  fourth.  &c. 
Cor.  If  quantities  are  in  geometrical  progression,  their 
differences  are  also  in  geometrical  progression. 

Thus  the  numbers  3,  9,     27,     81,       243,  &c. 

And  their  differences  0,  18,     54,     162,  &c.  are  in 

geometrical  progression. 

444.  Several  quantities  are  said  to  be  in  harmonicol  pro- 
gression, when,  of  any  three  which  are  contiguous  in  the  se- 
ries, the  first  is  to  the  last,  as  the  difference  between  the  two 
first,  to  the  difference  between  the  two  last.     See  art.  409. 

Thus  the  numbers  60,30,  20,  15,  12,  10,  are  in  harmoni- 
cal  progression. 

\For60:  20:: 60— 30:30-20,  And  20: 12:: 20- 15: 15-12, 
\And30: 15::  30-20:20— 15,  And  15  : 10::15-12:12-10 


,    /  •    ,  •" 

'.••fy         f-''/#t     ' 


SECTION 


INFINITES  AND  INFINITESIMALS  * 


ART  445  T^^  word  infinite  is  used  in  different  senses. 
The  ambiguity  of  the  term  lias  been  the  oc- 
casion of  much  perplexity.  It  has  even  led  to  the  abr-urd 
supposition,  that  propositions  directly  contradictory  to  each 
other  may  be  mathematically  demonstrated  These  appar- 
ent contradictions  are  owing  to  the  fact,  that  what  is  proved 
of  infinity,  when  understood  in  one  particular  manner,  is  of- 
ten thought  to  be  true  also,  when  the  term  has  a  very  differ- 
ent signification.  The  two  meanings  are  insensibly  shifted, 
the  one  for  the  other,  so  that  the  proposition  which  is  really 
demonstrated,  is  exchanged  for  another  which  is  false  and 
absurd.  To  prevent  mistakes  of  this  nature,  it  is  important 
that  the  different  meanings  be  aurefully  distinguished  from 
each  other. 

446.  INFINITE,  in  the  highest,  and  perhaps  the  most 
proper  sense  of  the  word,  is  that  which  is  so  great,  that  no- 
thing can  be  added  to  it,  or  supposed  to  be  added. 

In  this  sense,  it  is  frequently  used,  in  speaking  of  moral 
and  metaphysical  subjects.  Thus,  by  infinite  wisdom  is 
meant  that  which  will  not  admit  of  the  least  addition.  Infi- 
nite power  is  that  which  cannot  possibly  be  increased,  even 
in  supposition.  This  meaning  of  infinity  is  not  applicable  to 
the  mathematics.  That  which  is  the  subject  of  the  mathe- 
matics is  quantity ;  (Art.  1.)  such  quantity  as  may  be  con- 
ceived by  the  human  mind.  But  no  idea  can  be  formed  of 
a  quantity  so  great  that  nothing  can  be  supposed  to  be  added 
to  it.  In  this  sense,  an  infinite  number  is  inconceivable.  We 
may  increase  a  number  by  continual  addition,  till  we  obtain 
one  that  shall  exceed  any  limits  which  we  please  to  assign. 
By  this,  however,  we  do  not  arrive  at  a  number  to  which 

*  Locke's  Essays,  Book  2.  Chap.  17.  Berkley's  Analyst.  Prc-ftce 
to  Maclaurin's  Fluxions.  Newton's  Princip.  Saurider^on's  Algebra, 
Art,  336.  Mansfield's  Essays,  Emerson's  Algebra,  Prob.  73. 


MATHEMATICAL  INFINITY.  221 

nothing  can  be  added ;  but  only  at  one  that  is  beyond  any 
limits  which  we  Have  hitherto  set.  Farther  additions  may 
be  made  to  it,  with  the  same  ease,  as  those  by  which  it  has 
already  been  increased  so  far.  It  is  therefore  not  infinite,  in 
the  sense  in  which  the  term  has  now  been  explained.  It  is 
absurd  to  speak  of  the  greatest  possible  number.  No  number 
can  be  imagined  so  great,  as  not  to  admit  of  being  made 
greater.  We  must  therefore  look  for  another  meaning  of  in- 
finity ,Jbefore  we  can  apply  it,  with  propriety,  to  the  mathe- 
matics. 

447.  A  mathematical  quantity  is  said  to  be  infinite,,  when  it  it 
supposed  to  be  increased  beyond  any  determinate  limits . 

By  determinate  limits  are  meant  such  as  can  be  distinctly 
stated.*  In  this  sense,  the  natural  series  of  numbers  J ,  2,  3, 
4,  5,  &c.  may  be  said  to  be  infinite.  For,  if  any  number  be 
mentioned  ever  so  great,  another  onay  be  supposed  still 
greater. 

The  two  significations  of  the  word  infinite  are  liable  to  be 
confounded,  because  they  are  in  several  points  of  view  the 
same.  The  higher  meaning  includes  the  lower.  That  which 
is  so  great  as  to  admit  of  no  addition,  must  be  beyond  any 
determinate  limits.  But  the  lower  does  not  necessarily  im- 
ply the  higher.  Though  number  is  capable  of  being  increas- 
ed beyond  any  specified  limits;  it  will  not  follow,  that  a  num- 
ber can  be  found  to  which  no  farther  additions  can  be  made. 
The  two  infinites  agree  in  this,  that,  according  to  each,  the 
things  spoken  of  are  great  beyond  calculation.  But  they 
differ  widely  in  another  respect.  To  the  one,  nothing  can 
be  added.  To  the  other,  additions  can  be  made  at  pleas- 
ure. 

448.  In  the  mathematical  sense  of  the  term,  there  is  no 
absurdity  in  supposing  one  infinite  greater  than  another. 

We  may  conceive  the  numbers         2222222  &c. 
and  4  4  4  4  4  4  4  fcc. 

to  be  each  extended  so  far  as  to  reach  round  the  globe,  or 
to  the  most  distant  visible  star,  or  beyond  any  greater  boun- 
dary which  can  be  mentioned.  But,  if  the  two  series  be 
equally  extended,  the  amouot  of  the  one  will  be  twice  a* 
great  as  the  other,  though  both  be  infinite. 

So,  if  the  series     a-f   a2-f   a3-f-  a4-f   a 
and 

*  See  Note  L. 


ALGEBRA. 

"be  extended  together  beyond  any  specified  limits,  one  wilt 
be  nine  times  as  great  as  the  other.  But  it  would  be  absurd 
to  suppose  one  quantity  greater  than  another,  if  the  latter 
•wore  already  so  great  that  nothing  could  be  added  to  it. 

449.  An  infinite  number  of  terms  must  not  be  mistaken  for 
an  infinite  quantity.  The  terms  may  bo  extended  beyond 
any  given  limits,  when  the  amount  of  die  whole  is  a  finite 
quantity,  and  even  a  small  one.  If  we  take  half  of  a  unit  ; 
then  half  of  the  remainder  ;  half  of  the  remaining 
»ve  shall  luve  the  series 


&C. 

in  which  each  succeeding  term  is  half  of  the  preceding  one. 
Let  the  progression  be  continued  ever  so  far,  the  sum  of  all 
the  terms  can  never  exceed  a  unit.  For,  by  the  supposition, 
there  is  still  a  remainder  equal  to  the  last  term.  And  this 
remainder  must  be  added,  before  the  amount  of  the  whole 
can  be  equal  to  a  uait 

So  f  +  S  +  I  +  A  +  T&  +  V*  &e-  can  never  -exceed  8. 

450.  When  a  quantity  is  diminished  till  it  becomes  LESS  than 
any  determinate  quantity,  it  is  called  an  INFINITESIMAL. 

Thus,  in  the  scries  of  fractions  jv,  ||F,  j^\t,  j^k^toc. 
a  unit  is  first  divided  into  ten  parts,  then  into  a  hundred,  a 
thousand,  Sec.  One  of  these  parts  in  each  succeeding  term, 
is  ten  times  less  than  in  the  preceding.  If  then  the  progres- 
sion be  continued,  a  portion  of  a  unit  may  be  obtained  less 
than  any  specified  quantity.  This  is  an  infinitesimal,  and,  in 
mathematical  language,  is  said  to  be  infinitely  small.  By  this, 
however,  we  are  not  to  understand,  that  it  can  not  be  made 
less.  The  same  process  that  has  reduced  it  below  any  limit 
which  we  have  yet  specified,  may  foe  continued,  so  as  to  di- 
minish it  still  more.  And  however  far  the  progression  may 
be  carried,  we  shall  never  arrive  at  a  point  where  we  must 
necessarily  stop. 

451.  In  the  sense  now  explained,  mathematical  quantity 
iaay  be  said  to  be  infinitely  divisible;  that  is,  it  may  be  sup"- 
poseJ  to  be  so  divided,  that  the  parts  shall  be  less  than  any 
determinate  quantity,  and  the  nffmber  of  parts  greater  than 
any  given  number. 

In  the  series  -^  7|T,  TTViri  lff|T>T,  &c.  a  unit  is  divided 
into  a  greater  and  greater  number  of  parts,  till  they  become 
infinitesimals,  and  the  number  of  them  infinite,  that  is,  such  a 
number  as  exceeds  anv  ghcn  number.  But  this  does  not 


MATHEMATICAL  INFINITY.  2235 

prove  that  we  can  ever  arrive  at  a  division  in  which  the  parts- 
shall  be  the  least  possible,  or  the  number  of  parts  the  greatest 
possible. 

452.  One  infinitesimal  may  be  less  than  another. 


The  series  T67,  Tfo,  TTaTT, 
And  -3 


may  be  carried  on  together,  till  the  last  term  in  each  be- 
comes infinitely  small ;  and  yet  one  of  these  terms  will  be 
only  half  as  great  as  the  other.  For,  the  denominators  be- 
ing the  same,  the  fractions  will  be  as  their  numerators,  (Art, 
360.  cor.  2.)  that  is,  as  G :  3,  or  2 : 1. 

Two  quantities  may  also  be  divided,  each  into  an  infinite 
number  of  parts,  using  the  term  infinite  in  the  mathematical 
sense,  and  yet  the  parts  of  one  be  more  numerous,  than  those 
of  the  other. 


The  series 
And 


:\ 


may  both  be  infinitely  extended ;  and  yet  a  unit  in  the  last 
series,  is  divided  into  four  times  as  many  parts  as  in  the  first* 
But  if,  by  an  infinite  number  of  parts  were  meant  such  a  num- 
ber as  could  not  be  increased,  it  would  be  absurd  to  suppose 
$ie  divisions  of  any  quantity  to  be  still  more  numerous.* 

453.  For  all  practical  purposes,  an  infinitesimal  may  be 
considered  as  absolutely  nothing.  As  it  is  less  than  any 
determinate  quantity,  it  is  lost  even  in  numerical  calculations. 
In  algebraic  processes,  a  term  is  often  rejected  as  of  no  val- 
ue, because  it  is  infinitely  small. 

It  is  frequently  expedient  to  admit  into  a  calculation  a 
small  errour,  or  what  is  suspected  to  be  an  errour.  It  may 
be  difficult  either  to  avoid' the  objectionable  part,  or  to  ascer- 
tain its  exact  vahiej  or  even  to  determine,  without  a  long  and 
tedious  process,  whether  it  is  really  an  errour  or  not.  But  if 
it  can  be  shown  to  be  infinitely  small,  it  is  of  no  account  in 
practice,  and  may  be  retained  or  rejected  at  pleasure. 

It  is  impossible  to  find  a  decimal  which  shall  be  exactly 
equal  to  the  vulgar  fraction  -|.  Dividing  the  numerator  by 
the  denominator,  we  obtain,  in  the  first  place  T\.  This  is 
»*arly  equal  to  |.  But  ^30  is  nearer,  -?^\  still  nearer,  &cv 

*  See  Note  M, 


23*  ALGEBRA. 

The  errour,  in  the  first  instance,  is 


In  the  same  manner  it  may  be  shown,  that 

the  difference  between   \  f  ancj  '3*'  IS.  ** 
I  ±  and  .333,  is 


If  the  decimal  be  supposed  to  be  extended  beyond  any 
assignable  limit,  the  difference  still  remaining  will  be  infinite- 
ly small.  As  this  errour  is  less  than  any  given  quantity,  it  is 
of  no  account,  and  may  be  considered  in  calulatiun  as  no- 
thing. 

454.  From  the  preceding  example  it  will  be  seen,  that  a 
quantity  may  be  continually  coming  nearer  to  another,  and 
yet  never  reach  it.    The  decimal  0.3333333  &c.  by  repeated 
additions  on  the  right,  may  be  made  to  approximate  contin- 
ually to  •£,  but  can  never  exactly  equal  it.     A  difference  will 

ays  remain,  though  it  may  become  infinitely  small. 

455.  Though  an  infinitesimal  is  of  no  account  of  itself,  yet 
its  effect  on  other  quantities  is  not  always  to  be  disregarded. 

When  it  is  a  factor  or  a  divisor,  it  may  have  an  important 
influence.  It  is  necessary,  therefore,  to  attend  to  the  rela- 
tions which  infinites,  infinitesimals,  and  finite  quantities  have 
to  each  other.  As  an  infinitesimal  is  less  than  any  assignable 
quantity,  as  it  is  next  to  nothing,  and,  in  practice,  may  be 
considered  as  nothing,  it  is  frequently  represented  by  0. 

An  infinite  quantity  is  expressed  by  the  character  00  . 

456.  As  an  infinite  quantity  is  incomparably  greater  than 
m  finite  one,  the  alteration  of  the  former,  by  an  addition  or 
subtraction  of  the  latter,  may  be  disregarded  in  calculation. 
A  single  grain  of  sand  is  greater  in  comparison  with  the 
whole  earth,  than  any  finite  quantity  in  comparison  with  one 
which  is  infinite.      If  therefore  infinite  and  finite   quantities 
are  connected  by  the  sign  -f  or  —  ,  the  latter  may  be  reject- 
ed as  of  no  comparative  value.      For  the  same  reason,  if  fi- 
nite quantities  and  infinitesimals  are  connected  by  -f-  or  —  , 
the  latter  may  be  expunged. 

457.  But  if  an  infinite  quantity  be  multiplied  by  one  which 
is  finite,  it  will  be  as  many  times  increased,  as  any  other 
quantity  would,  by  the  same  multiplier. 

If  the  infinite  series  2  2  2  2  2  2  Sec.    be  multiplied  by  4, 
The  product  will  be  888888  &cc.  four  times  as  great 
ns  the  multiplicand.     See  art.  4-48. 


MATHEMATICAL  INFINITY.  C2S 

458.  And  if  an  infinite  quantity  be  divided  by  a  finite 
•quantity,  it  will  be  altered  in  the  same  manner  as  any  other 
quantity. 

If  the  infinite  series    6GG66666  &c.  be  divided  by  2, 
The  quotient  will  be  33333333  &c.  half  as  great  as  the 
dividend. 

459.  If  a  finite  quantity  be  multiplied  t>y  an  infinitesimal, 
.the  product  will  be  an  infinitesimal ;  that  is,  putting  z  for  a 
^finite  quantity,  and  0  for  an  infinitesimal,  (Art.  455.) 


If  the  multiplier  were  a  unit,  the  product  would  be  equal 
to  the  multiplicand.  (Art.  90.)  If  the  multiplier  is  less  than 
a  unit,  the  product  is  proportionally  Jess.  If  then  the  mul- 
tiplier is  infinitely  less  than  a  unit,  the  product  must  be  infi- 
nitely less  than  the  multiplicand,  that  is,  it  must  be  an  infi- 
nitesimal. Or,  if  an  infinitesimal  be  considered  as  abso- 
lutely nothing,  then  the  product  of  z  into  nothing  is  nothing. 
(Art.  112.) 

•     460.    Op  the  other  hand,  if  a  finite  quantity  be   divi- 
ded by  ai/  infinitesimal,  the  quotient  will  be  infinite. 

!=*• 

For,  the  less  the  divisor,  the  greater  the  quotient.  If 
then  the  divisor  be  infinitely  small,  the  quotient  will  be  infi- 
nitely great.  In  other  words,  an  infinitesimal  is  contained 
an  infinite  number  of  times  in  a  finite  quantity.  This  may, 
at  first,  appear  paradoxical.  But  it  is  evident,  that  the  quo- 
tient must  increase,  as  the  divisor  is  diminished. 

Thus  6-4-3=2,  6  -f-  0.03 = 200, 

6-^-0.3=20,  £4- 0.003  =2000,  &c. 

If  then  the  divisor  be  roduced,  so  as  to  become  less  than 
any  assignable  quantity,  the  quotient  must  be  greater  than 
any  assignable  quantity. 

4€1.  If  a  finite  quantity  be  divided  by  an  infinite  quantity, 
the  quotient  will  be  an  infinitesimal. 


oc 


-0. 


For,  the  greater  the  divisor,  the  less  the  quotient      If 
Ee 


226  ALGEBRA. 

then,  while  the  dividend  is  finite,  the  divisor  be  infinitely 
great,  the  quotient  will  be  infinitely  small. 

It  must  not  be  forgotten,  that  the  expressions  infinitely 
#reerf,  and  infinitely  small  are,  all  along,  to  be  understood  iu 
t^e  mathematical  sense,  according  to  the  definitions  iu  aits. 
447,  and  450.  I 


SECTION    XVI. 


DIVISION  BY  COMPOUND  DIVISORS. 


A  462  T^  t^le  sect*on  on  division,  the  case  in  which 
the  divisor  is  a  compound  quantity  was  omit- 
ted, because  the  operation,  in  most  instances,  requires  some 
knowledge  of  the  nature  of  powers  ;  a  subject  which  had  not 
been  previously  explained. 

Division  by  a  compound  divisor  is  performed  by  the  .fol- 
lowing rule,  which  is  substantially  the  same,  as  the  rule  for 
division  in  arithmetiq : 

To  obtain  the  first  term  o£  the  quotient,  divide  the  first 
term  of  the  dividend,  by  the  first  term  of  the  divisor  :* 

Multiply  the  whole  divisor,  by  the  term  placed  in  the  quo- 
tient ;  subtract  the  product  from  a  part  of  the  dividend ;  and 
to  the  remainder  bring-  down  as  many  of  the  following  terms, 
as  shall  be  necessary  to  continue  the  operation : 

Divide  again  by  the  first  term  of  the  divisor,  and  proceed 
as  before,  tili  aH  the  terms  of  the  dividend  are  brought  down. 

Ex.  1.  Divide  ac+bc+ad+bd,  by  a +5. 

a + b}ac + be 4- ad+ bd(c + d 

ac+bc,  the  first  subtrahend. 


ad-\-bd 

ad+bd,  the  second  subtrahend. 


Here  ac,  the  first  term  of  the  dividend,  is  divided  by  <r, 
the  first  term  of  the  divisor,  (Art.  116.)  which  gives  c  for  the 
first  term  of  the  quotient.  Multiplying  the  whole  divisor  by 
this,  we  have  ac+bc  to  be  subtracted  from  the  two  first  term? 
of  the  dividend.  The  two  remaining  terms  are  then  brought 

*  Sec  Note  N. 


228  ALGEBRA 

down,  and  the  first  of  them  is  divided  by  the  first  term  of 
the  divisor,  as  before.  This  gives  d  for  the  second  term  of 
the  quotient.  Then  multiplying  the  divisor  by  </,  we  have 
ad+bd  to  be  subtracted,  which  exhausts  the  whole  dividend, 
without  leaving  any  remainder. 

The  rule  is  founded  on  this  principle,  that  the  product  of 
the  divisor  into  the  several  parts  of  the  quotient,  is  equal  to- 
the  dividend.  (Art.  115.)  Now  by  the  operation,  the  pro- 
duct of  the  divisor  into  the  first  term  of  the  quotient  is  sub- 
tracted from  the  dividend;  then  the  product  of  the  divisor 
into  \he  second  term  of  the  quotient;  and  so  on,  till  the  pro- 
duct of  the  divisor  into  each  term  of  the  quotient,  that  is, 
the  product  of  the  divisor  into  the  whole  quotient,  (Art.  100.) 
is  taken  from  the  dividend.  If  there  is  no  remainder,  it  is 
evident  that  this  product  is  equal  to  the  dividend.  If  there 
is  a  remainder,  the  product  of  the  divisor  and  quotient  is 
equal  to  the  whole  of  the  dividend  except  the  remainder. 
And  this  remainder  is  not?  included  in  the  parts  subtracted 
from  the  dividend,  by  operating  according  to  the  rule. 

463.  Before  beginning  to  divide,  it  will  generally  be  ex- 
pedient to  make  some  preparation  in  the  arrangement  of  the 
MMttt. 

The  letter  which  is  in  the  first  term  of  the  divisor,  should 
be  in  the  first  term  of  the  dividend  also.  And  the  powers  of 
this  letter  should  be  arranged  in  order,  both  in  the  divisor 
and  in  the  dividend  ;  the  highest  power  standing  first,  the 
next  highest  next,  and  so  on. 


Ex.  2.  Divide  2a26+i3+2oZ>*+as,  Byaa4-&*+«k 

Here  if  AVG  take  a2  for  tile  first  term  of  the  divisor,  the 
ether  terms  should  be  arranged  according  to  the  powers  o& 
a,  thus; 


ab 


In  these  operations,  particular  care  will  be  necessary  in 
the  management  of  negative  quantities.    Constant  attention 


DIVISION.  229 

jtfust  be  paid  to  the  rules  for  the  signs  in  subtraction,  multi- 
plication and  division.  (Arts.  82,  105,  123.) 


Ex.  3".  Divide  2o#—  2«*,r—  3a*xy+8a*x+axy—  «y,  by 
2a—y. 

If  the  terms  be  arranged  according  to  the  powers  of  a, 
they  will  stand  thus  ; 

2a—    6a*x 


*      — 


+2ar— xy 


464.  In  multiplication,  some  of  the  terms,  by  balancing 
«ach  other,  may  be  lost  in  the  product.  (Art.  110.)  These 
may  re-appear  in  division,  so  as  to  present  terms,  in  the  course 
of  the  process,  different  from  any  which  are  in  the  dividend* 

Ex.   4. 
'-f  x3(a2—  or-4-a;a 


*  — 


ax*+x* 


Ex.    5. 


236  ALGEBRA. 

If  the  learner  will  take  the  trouble  to  multiply  the  quo- 
tient into  the  divisor,  in  the  two  last  examples,  he  will  find, 
in  the  partial  products,  the  several  terms  which  appear  in  the 
process  of  dividing.  But  most  of  them,  by  balancing  each 
other,  are  lost  hi  the  general  product. 

Ex.    6. 


*      3ac+3c 
3ac+3c 


Ex.    7. 

a-f&— c(a+b— c— ax— bx-\-cx(l—  x 
a  +  b—c 


*    *    *—ax  —  bx-\-cx 
—  ax—bx+cx 


Ex.  8.  Divide  2a*-13a3x+lla*x* -Sax* +2**,    by 
2az—  ax+x*.  -Quotient,     a3—  6a#-f2r2. 

465.  When  there  is  a  remainder  after  all  the  terms  of  the 
dividend  have  been  brought  down,  this  may  be  placed  over 
the  divisor  and  added  to  the  quotient,  as  in  arithmetic. 

Ex.    9. 

a+b}ac-irbc+ad 
ac+bc 


*      ad+bd 
ad+bd 

*       * 


DIVISION. 
Ex.  10. 


— h)ad— ah+bd— 
ad— ah 


*      *      bd-bh 
bd-bh 


It  is  evident  that  a+b  is  the  quotient  belonging  to  the 
whole  of  the  dividend,  excepting  the  remainder  y.     (Art. 

M 
562.)     And  j_i  is  the  quotient  belonging  to  this  remainder. 

(Art.  124.) 

Ex.  11.  Divide  Gax+Zicy—  3a6—  by+3ac+cy+h,    by 
3a+y.  Quotient.  2r—  ft+c-f 


Ex.  12.  Divide  a26-3a2+2a5-6a-46+22,  by  6-3. 

10 
Quotient.   a2+2a—  4+r^g' 

Ex.  13.     See  art.  283. 


Ex.  14. 


ar^y+ry 
ar^y+ry 


i  .  ;_ 


/  . 
/ 


252  ALGEBRA. 

466.  A  regular  series  of  quotients  is  obtained,  by  dividing 
sthe  difference  of  the  powers  of  two  quantities,  by  the  diflej> 
cnce  of  the  quantities.  Thus, 

(ys-a2)-r(y-a)=y-fa, 


+«2y 


Here  it  will  be  seen,  that  the  index  of  y,  in  tire  first  term 
of  the  quotient,  is  less  by  1,  than  in  the  dividend;  and  that 
it  decreases  by  1,  from  the  first  term  to  the  last  but  one  : 

While  the  index  of  a  increases  by  1,  from  the  second 
term  to  the  last,  where  it  is  less  by  1,  than  in  the  dividend. 

This  may  be  expressed  in  a  general  formula,  thus, 


To  demonstrate  this,  we  hare  only  to  multiply  the  quo- 
tient into  the  divisor.  (Art.  115.) 

All  the  terms  except  two,  in  the  partial  products,  will  be 
balanced  by  each  other  ;  and  will  leave  the  general  product 
the  same  as  the  dividend. 

Mult.  y*+ay*+azyz+a3y+a* 
Into    y  —  a 


y4  +«y4  +«V  +a'y2  -f  «*y 

—  ay*  —  a2y3  —  asy2  —  a*y  —  a5 


Product,  y5 


*   -5 


So  mult.  yW!-1+a/1-2+a2yw-5 ....  +am-8y  -fa1"-* 
Into         y—a 

+am~2y2  +am~ly 

am-2y  2  _  am-ly  _ 

ProdL       ym     *  -am. 

\ 


SfcCTION 


INVOLUTION  AND  EXPANSION  OF  BINOMIALS.* 


A  4.87  TPHE  manner  in  which  a  binomial,  as  well  as 
any  other  compound  quantity,  may  be  invol- 
ved by  repeated  multiplications,  has  been  shown  in  the  sec- 
tion on  powers.  (Art.  213.)  But  when  a  high  power  is  re- 
quired, the  operation  becomes  long  and  tedious. 

This  has  led  mathematicians  to  seek  for  some  general 
principle,  by  which  the  involution  may  be  more  easily  and 
expeditiously  performed;  We  are  chiefly  indebted  to  Sir 
Isaac  Newton  for  the  method  which  is  now  in  common  use. 
It  is  founded  on  what  is  called  the  Binomial  Theorem,  the  in- 
vention of  which  was  deemed  of  such  importance  to  mathe- 
matical investigation,  that  it  is  engraved  on  his  monument  in 
Westminster  Abbey. 

468.  If  the  binomial  root  be  «  +  &,  we  may  obtain,  by  mul* 
tiplication,  the  following  powers.  (Art.  213.) 


By  attending  to  this  series  of  powers,  we  shall  find,  that 
the  exponents  preserve  an  invariable  order  through  the  whole- 
This  will  be  very  obvious,  if  we  take  the  exponents  by  them- 
selves, unconnected  with  the  letters  to  which"  they  belong. 

*  Simpson's  Algebra,  Sec.  15.  Simpson's  Flaxions,  Art.  39.  Etr- 
ler's  Algebra,  Sec.  2.  Chap.  10.  Manning's  Algebra.  Saunderson's- 
Algebra,  Art.  380.  Vincc's  Fluxions,  Art.  33.  Waring's  Med.  Anal. 
p.  415.  Lacroix's  Algebra,  Art.  133.  Do.  Comp.  Art.  70.  Loud. 
Phil.  Trans.  1795. 

Ff 


4  ALGEBRA, 

• 

In  the  square,  the  exponents          °r  ?  *  C  n  i    o 

(  of  b  are  0,  1 ,  2 

fn  the  cube,  the  exponents  °r  ?  a   !  J:»  '  ' 

(of  6  are  0,1, 2,3 

In  the  4th  power,  the  exponents  \  °J  ?  are  J?'H'? 

(  ofo  are  0,1,2,3,4 


are  0,1,2,3,4,5,  &c. 
Here  it  will  be  seen,  at  once,  that  the  exponents,  of  a  in 
the  first  term,  and  of  b  in  the  /as?,  are  each  equal  to  the  in- 
dex of  the  power;  and  that  the  sum-of  the  exponents  of  the 
two  letters  is  in. every  term  the  same.  Thus  in  the  fifth 
power, 

(  fn  the  first  term,  is  5  +  0=5 
The  sum  of  the  exponents  <  in  the  second,        4-f  1  =5 

f  ia  the  third,  3+2=5,&c. 

It  is  farther  to  be  observed,  that  the  exponents  of  a  regu- 
larly decrease  to  0,  and  that  the  exponents  of  b  increase  from.' 
0.  That  this  will  universally  be  the  case,  to  whatever  ex- 
tent the  involution  may  be  carried,  will  be  evident,  if  we 
consider,  that,  in  raising  from  any  power  to  the  next,  each 
term  is  multiplied  both  by  a  and  by  b. 

Thus  (o+6)2=aa  +2ai-f4* 
Mult,  by  «+& 

[of  a  in  each  term. 

a3  +2a2Z»+a62,  Here  1  is  added  to  the  exp. 
a2b  +  2ab2+b3,  Here  1  is  added  to  the 

Texp.  of/)  in  each  term. 
3 


If  the  exponents,  before  the  multiplication,  increase  and 
decrease  by  1,  and  if  the  multiplication  adds  1  to  each,  it  i* 
evident  they  must  still  increase  and  decrease  in  the  same 
manner  as  before. 

469.  If  then  a-\-b  be  raised  to  a  power  whose  exponent 
is  n, 

The  exp's  of  a  will  he      n,  n— 1,  n — 2, ....   2,       1,       0 ; 
And  the  exp'sof  b  will  be  0,     1,       2,      ....n— 2,n  — 1,  n. 

The  terms  in  which  a  power  is  expressed,  consist  of  the 
Icitsrs  with  their  exponents,  and  the  co-ejjicicnts.  Setting 


INVOLUTION  OF  BINOMIALS.  23K 

• 

aside  the  co-efficients  for  the  present,  we  can  determine, 
from  the  preceding  observations,the  letters  and  exponents  of 
any  power  whatever. 

Thus  the  8th  power  of  a-f  5,  when  written  without  the 
co-efficients,  is 


And  the  nth  power  of  a+6  is, 


470.  The  number  of  terms  is  greater  by  1  ,  than  the  index 
of  the  power.     For,  if  the  index  of  the  power  is  n,  a  has,  in 
different  terms,  every  index  from  n  down  to  1  ;  and  there  is 
one  additional  term  which  contains  only  b.    Thus,  ' 

The  square  has  2  terms,         The  4th  power,  5, 
The  cube  4,  The  5th  power,  6,  &c. 

471.  The  next  step  is  to  find  the  co-efficients.      This  pan 
of  the  subject  is  more  complicated. 

In  the  series  of  powers  at  the  beginning  of  art.  468,  the 
co-efficients,  taken  separate  from  the  letters,  are  as  follows; 

In  the  square,  1,  2,  1,       whose  sum  is  4=2% 

In  the  cube,  ],  3,  3,  1,                            8=23, 

In  the  4th  power,  1,  4,  G.  4,  1,                       16  =24, 

In  the  5th  power,  1,5,10,10,5,1,                    .32=25. 

The  order  which  these  co-efficients  t>bserve;  is  not  obvious, 
like  that  of  the  exponents,  upon  a  bare  inspection.  But 
they  will  be  found  on  examination  to  be  all  subject  to  the 
following  law  $*H 

472.  The  co-efficient  of  the  first  term  is  1  ;  that  of  the 
second  is  equal  to  the  index  of  tire  power  ;  and  universally, 
if  the  -co-efficient  of  any  term  be  multiplied  by  the  ind£x  oi 
the  leading  quantity  in  that  term,  and  divided  by  the  index  of 
the  following  quantity  increased  by  1,  it  will  give  the  co-effi- 
cient of  the  succeeding  term.* 

Of  the  two  letters  in  a  term,  the  first  is  called  the  leading 
quantity,  and  the  other,  the  following  quantity.  In  the  ex- 
amples which  have  been  given  in  this  section,  «  is  the  lead- 
Jng  quantity,  and  b  the  following  quantity. 

It  may  frequently  be  convenient  to  represent  the  co-efil- 
/cients,  in  the  several  terms,  by  the  capital  letters.  «#,/?,  O.  &c 

*  See  Not*  O. 


236  ALGEBRA. 

The  nth  power  of  a  +  l,  without  the  co-efficients,  is 
aa+an-J&+a"-2£*+an-363+a"-46*,  &tc.  (Art.  469.) 

And  the  co-efficients  are, 
^=n,  the  co-efficient  of  the  second  term  ; 


=  n  x  ~~2~»  °    ^e  ****  terra  > 

?i  —  l     n—  2 
C  =  n  X  ~2~~~x~3~>  °f  tue  fourth  term; 

n—  1     n—  2    n—  3 
J/  =  n  X  —  g—  X  -"g—  X  —  £-,  of  the  fifth  term,  &c. 

The  regular  manner  in  which  these  co-efficients  are  de- 
rived one  from  another,  will  be  readily  perceived. 

473.  3y  recurring  to  the  numbers  in  art.  471  ,  it  will  be  seen, 
that  the  co-efficients  first  increase,  and  then  decrease  at  the 
same  rate  ;  so  that  they  are  equal,  in  the  first  term  and  the 
last,  in  the  second  and  last  but  one,  in  the  third  and  last  but 
two  ;  and,  universally,  in  any  two  terms  equally  distant  from 
the  extremes.    The  reason  of  this  is,  that  («  +  &)"  is  the  same 
as  (6+c)n;  and  if  the  order  of  the  terms  in  the  binomial 
root  be  changed,  the  whole  series  of  terms  in  the  power  will 
be  inverted. 

It  is  sufficient,  then,  to  find  the  co-cfiicients  of  half  the 
terms.     These  repeated,  will  serve  for  the  whole. 

474.  In  any  power  of  (a  +5),  the  sum  of  the  co-efficients 
is  equal  to  the  number  2  raised  to  that  power.      See  the  list 
of  co-efficients  in  art.  471.     The  reason  of  this  is,  that,  ac- 
cording to  the  rules  of  multiplication,  when  any   quantity  is 
involved,   the  letters  are  multiplied  into  each  other,  and  the 
co-efficients  into  each  other.      Now  the  co-efficients  of  a  +  b 
being  1  +  1  =2,  if  these  be  involved,  a  series  of  the  powers 
of  2  will  be  produced. 

Multiplying  1+1  or  2 

Into  1  +  1  2 

The  square  is  1+2+1  or  4=  the  square  of  2 

Mult,  again       1  +  1  2 

The  cube  is     1+3+3+1         or  8  =the  cube  of  2,  fce, 


INVOLUTION  OF  BINOMIALS.  237 

475.  The  principles  which  have  now  been  explained  may 
mostly  be  comprised  in  the  following  general  theorem,  called 

THE  BINOMIAL  THEOREM. 

The  index  of  the  leading  quantity  of  the  power  of  a  binomi- 
inial,  begins  in  the  first  term  with  the  index  of  the  power,  and 
decreases  regularly  by  1.  The  index  of  the  following  quantity 
begins  with  1  in  the  second  term,  and  increases  regularly  by  1. 
(Art.  468.) 

The  co-efficient  of  the  first  term  is  \  ;  that  of  the  second  is 
equal  to  the  index  of  the  power  ;  and  universally,  if  the  co-effi- 
cient of  any  term  be  multiplied  by  the  index  of  the  leading 
quantity  in  that  term,  and  divided  by  the  index  of  the  following 
quantity  increased  by  1  ,  it  will  give  the  co-efficient  of  the  suc- 
ceeding term.  (Art.  472.) 

In  algebraic  characters,  the  theorem  is. 

(a  +  b}*  =  an  +  n  X  an~lb  +  n  X  —^-an~2b  2  ,  SEC. 

It  is  here  supposed,  that  the  terms  of  the  binomial  have 
no  other  co-efficients  or  exponents  than  1.  Other  binomials 
may  be  reduced  to  this  form  by  substitution. 

Ex.  1.  What  is  the  6th  power  of  x-\-y? 

The  terms  without  the  co-efficients,  are 

a;8,  xsy,  x*y2,  x3y5,  xzy*,  zys,  y*. 
And  the  co-efficients  are 

6x5     15x4     20x3 


that  is,     1,     6,     15,         20,          15,      6,     1. 

Prefixing  these  to  the  several  terms,  we  have  the  power 
required  j 


238  ALGEBRA. 

3.  What  is  the  nth  power  of  b+y  ? 
Ans.  bn  -\-Al 


That  is,  supplying  the  co-efficients  which  arc  here  repre- 
sented by  A,  B,  C,  &c.  (Art.  472.) 

I*  +n  X  ln- 

476.  A  residual  quantity  raay  be  involved  in  the  same 
•manner,  without  any  variation,  except  in  the  signs.  By  re- 
peated multiplications,  as  in  art.  213,  we  obtain  the  follow- 
ing powers  of  (a—  6). 


By  comparing  these  with  the  like  powers  of  (a+ft)  in  art. 
468,  it  will  be  seen,  that  there  is  no  difference,  except  in 
the  signs.  There,  all  the  terms  are  positive.  Here,  the 
ierms  which  contain  the  odd  powers  of  b  are  negative.  See 
art.  218. 

The  sixth  power  of  (•£—  y)  is 


The  nth  power  of  (a—  b]  is 
a"  —Ad 


477.  When  one  of  the  terms  of  a  binomial  is  a  unit,  it  is 

generally  omitted  in  the  power,  except  jn  the  first  or  last 

fterm;  because  every  power  of  1  isl,   (Art.  209.)    and  this, 

"when  it  is  a  factor,  has  no  effect  upon  the  quantity  with 

which  it  is  connected.  (Art.  90.) 

Thus  the  cube  of  (jc+1)  is      a:3  +3*3  x  l+3rx  !*  +  !*, 
Which  is  the  same  as  x3  +  3x2-\-3x-{-l. 

The  insertion  of  the  powers  of  1  is  of  no  use,  unless  it 
be  to  preserve  the  exponents  of  bofh  the  leading  and  the  fol- 
lowing quantity  in  each  term,  for  the  purpose  of  finding  the 
co-efficients.  But  this  will  be  unnecessary,  if  we  bear  in 
Uiind,  that  the  sum  of  the  two  exponents,  in  each  term,  is 
equal  to  the  index  of  the  power.  (Art.  4C8.)  So  rhat,  if  we 


INVOLUTION  OF  BINOMIALS. 

Save  the  exponent  of  the  leading  quantity,  we  may  know- 
that  of  the  following  quantity,  and  v.  v. 

Ex.  1.  The  sixth  power  of   (1-  —  y)  is 

-20y3  " 


478.  From  the  comparatively  simple  manner  in  which  the 
power  is  expressedr  when  the  first  term  of  the  root  is  a  unit, 
is  suggested  the  expediency  of  reducing  other  binomials  to 
this  form. 

The  quotient  of  (a+x)  divided  by  ais((l  +  —  ).   This 

\  Cv  * 

multiplied  into  the  divisor,  is  equal  to  the  dividend;  that  is, 

/         x\  I        x\n 

z)=«X  \  1  +  ~~J  therefore    (a+x}n  =afl  X  I  1+—  J  -' 

\  Ci  '  *  Gf  • 


I  X\n 

By  expanding  the  factor  ^  1  -f  —  j   ,  we  have 


47&.  AVhen  the  index  of  the  power  to  which  any  binomi- 
al is  to  be  raised  is  a  positive  whole  number,  the  series  will  ter-~ 
minale.  The  number  of  terms  will  be  limited,  as  in  all  the 
preceding  examples. 

For,  as  the  index  of  the  leading  quantity  continually  de- 
creases by  1,  it  must,  in  the  end,  become  0,  and  then  the  se- 
jies  will  break  ofR 

Thus,  the  5  term  of  .the  fourth  power  of  a+x  is  x*,  or 
«°a?4,  a°  being  commonly  omitted,  because  it  is  equal  to  1.. 
(Art.  207.)  If  we  attempt  to  continue  the  series  farther, 
the  co-efficient  of  the  next  term,  according  to  the  rule,  will  be 
1x0 
—  —  =0.  (Art.  112.)  And  as  the  co-efficients  of  all  suc- 

ceeding terms  must  depend  on  this,  they  will  also  be  0. 

480.  If  the  index  of  the  proposed  power  is  negative,  this 
ean  never  become  0,  by  the  successive  subtractions  of  a  unit. 
The  series  will,  therefore,  never  terminate  ;  but,  like  many 
decimal  fractions,  may  be  continued  to  any  extent  that  5* 
desired. 


£10  ALGEBRA 

Ex.    Expand  into  a  series  /    .  -\2  ^ 
The  terms,  without  the  co-efficients,  are 

a"2,  a~3y,  a~*y2,  a~sy3,  ar*y*t-toc 

i    C  y  _  £ 

The  <ro-ef.  of  the  2d  term  is-2,  of  the  4th  -  -  ip—  =  -4, 

-2x-3  -4x-5 

of  the  3d,  --  £  -  =  +  3,     of  the  5th  -  ^  -  =4*5. 

The  series  then  is 
fl-2-2a-3y-{-3a-y  -4a~y  -f  5a-y  ,  &c. 

Here  the  law  of  the  progression  is  apparent  ;  the  co-effi- 
cients increase  regularly  by  1,  and  their  signs  are  alternately 
positive  and  negative. 

~48l.  The  Binomial  Theorem  is  of  great  utility,  not  only 

in  raising  powers,  but  particularly  in  finding  the  roots  of  bi- 

nomials.    A  root  may  be  expressed  in  the  same  manner  as  a 

power,  except  that  the  exponent  is,  in  the  one  case  an  inte- 

er,  in  the  other  a  fraction.    (Art.  245.)     Thus  (a+b)"  may 

e  either  a  power  or  a  root.  It  is  a  power  if  n=2,  but  a  root 

if  n=f 

482.  If  a  root  be  expanded  by  the  binomial  theorem,  the 
series  will  never  terminate.  A  series  produced  in  this  way 
terminates,  only  when  the  index  of  the  leading  quantity  be- 
comes equal  to  0,  so  as  to  destroy  the  co-efficients  of  the 
succeeding  terms.  (Art.  479.)  But,  according  to  the  theo- 
rem, the  difference  in  the  index,  between  one  term  and  the 
riext,  is  always  a  unit  ;  and  a  fraction,  though  it  may  change 
from  positive  to  negative,  can  not  become  exactly  equal  to  0, 
by  successive  subtractions  of  a  unit.  ,  Thu's,  if  the  index  in 
the  first  term  be  £,  it  will  be, 

In  the  2d,      £  -  1  =  —  |,     In  the  4th,  -£  —  1  =  —  f  , 
In  the  3d,  -|  -  1  =  —  |,     In  the  5th,  -  §  -  1  =  —  j,  &c. 

Ex.  What  is  the  square  root  of  (a-\-b)  ? 
The  terms,  without  the  co-efficients,  are 

«*,  a    *&,    a~~hz,  a~~*63,  a~H4,  See. 

The  co-efficient  of  the  2d  term  is  4--}, 

t  )(    _  J.  _  J^  y    _  1 

of  the  3d,—  -a"1  =  —  £,     of  the  4th,—  *-  —  1=  +  TV 


nd  the  series  is  a 


EXPANSION  OF  BINOMIALS.  241 

48i5.  The  binomial  theorem  may  also  be  applied  to  quan- 
tities consisting  of  more  than  two  terms.  By  substitution,  sev- 
eral terms  may  be  reduced  to  two,  and  when  the  compound 
expressions  are  restored,  such  of  them  as  have  exponents 
may  be  separately  expanded. 

Ex.  What  is  the  cube  of  a+b+c  ? 
Substituting  h  for  (b+c),  we  have  a+(b4-c)=a+k. 
And  by  the  theorem,  (a+h)3  =«3  +3a2A+3afta  +AS. 
That  is,  restoring  the  value  of  h, 

=as+3a*x(b+c)+3ax  (b+c)*  +(b+c)3. 


The  two  last  terms  contain  powers  of  (b+c)  ;  but  these 
jnay  be  separately  involved. 

' 
/, 

..     . 


>  l^-r  - 

__/» 

--  '      '  :  • 

rd^. 


:      > 
'       -^ 

IHt^^H^ 

r>{  i  I 

^<~  '      -&  \  — 

'"• 


< 


--//  ~2£  -VL  ££;•-  .'jJJ£-"+  ^  V. 


EVOLUTION  OF  COMPOUND  QUANTITIES. 


ART    481    HP^E  roots  of  compound  quantities  may  be  ex- 
•   tracted  by  the  following  general  rule  : 

After  arranging  the  terms  according  to  the  powers  of -one 
of  the  letters,  so  that  the  highest  power  shall  stand  first,  the 
next  highest  next,  &c. 

Take  the  root  of  the  first  term,  for  the  fast  term  of  the  re- 
quired root : 

Subtract  the  power  from  the  given  quantity,  and  divide  the 
first  term  of  the  remainder,  by  the  first  term  of  the  root  invol- 
ved to  the  next  inferiour  power,  and  multiplied  by  the  index  of 
the  given  power  ;f  the  quotient  ivill  be  the  next  term  of  the  root. 

Subtract  the  power  of  the  terms  already  found  from  the  giv- 
en quantity,  and,  using  the  same  divisor,  proceed  as  before. 

This  rule  verifies  itself.  For  the  root,  whenever  a  new 
term  is  added  to  it,  is  involved,  for  the  purpose  of  subtract- 
ing its  power  from  the  given  quantity ;  and  when  the  power 
is  equal  to  this  quantity,  it  is  evident  the  true  root  is  found. 

Ex.  1.  Extract  the  cube  root  of 

a«+3a5-3a4-lla3+6 
a6,  the  first  subtrahend. 


3a4)  *      3a6,  &c.  the  first  remainder. 

a6+3a4+3a4  +  as,  the  2d  subtrahend. 


3a4)  *       *    — 6a4,  Sic.  the  2d  remainder. 


f  By  the  given  power  is  meant  a  power  of  the  same  name  with  the 
required  root.  As  powers  and  roots  are  correlative,  any  quantity  it 
the  square  of  its  square  root,  the  cube  of  its  cube  root,  &c. . 


EVOLUTION.  .  243 

Here  a2,  the  cube  root  of  afl,  is  taken  for  the  first  term  of 
the  required  root.  The  power  a9  is  subtracted  from  the 
given  quantity.  For  a  divisor,  the  first  term  of  the  root  is 
squared,  that  is,  raised  to  the  next  inferiour  power,  and  mul- 
tiplied by  3,  the  index  of  the  given  power. 

By  this,  the  first  term  of  the  remainder  3a5 ,  fcc.  is  divided, 
and  the  quotient  a  is  added  to  the  root.  Then  a2  +a,  the 
part  of  the  root  now  found,  is  involved  to  the  cube,  for  the 
second  subtrahend,  which  is  subtracted  from  the  whole  of 
the  given  quantity.  The  first  term  of  the  remainder  — 6a4, 
&c.  is  divided  by  the  divisor  used  above,  and  the  quotient 
—2  is  added  to  the  root.  Lastly,  the  whole  root  is  involved 
to  the  cube,  and  the  power  is  found  to  be  exactly  equal  to 
the  given  quantity. 

It  is  not  necessary  to  write  the  remainders  at  length,  aSj  in 
dividing,  the  first  term  only  is  wanted. 

2.  Extract  the  fourth  root  of 

«4  +8as  +24a2  +  32a+  16(a-f2 


4a3(*      8a3,  &c. 

a4  +  8a3+24a*+32a-flG, 

3.  What  is  the  5th  root  of 

a5+5a4&  +  10a3&2+10a2£3+5a64  +  65? 

Ans.  a  -f-& 

4.  What  is  the  cube  root  of" 

a3-6a»6+12a6*-S63?  Ans.  a-Zb. 

5.  What  is  the  square  root  of 


4a)*     —  12a&,  &c 
4a* 


4a)  *  *  +16aA,  Stc. 


4a  2  -  I2ab  +  95  *  .  +  16a/i  - 


3244  ALGEBRA. 

In  finding  the  divisor  here,  the  term  2a  in  the  root  is  hot 
involved,  because  the  power  next  below  the  square  is  the 
first  power.  V^ 

485.  But  'the  square  root  is  more  commonly  extracted  by 
the  following  rule,  which  is  of  the  same  nature,  as  that 
which  is  used  ia  arithmetic. 

After  arranging  the  terms  according  to  the  powers  of  onft 
of  .tHe  letters,  take  the  root  of  the  first  term,  for  the  first 
term  of  the  required  root,  and  subtract  the  power  from  the 
given  quantity. 

Bring  down  two  other  terms  for  a  dividend.  Divide  by 
double  the  root  already  found,  and  add  the  quotient,  both  to 
the  root,  and  to  the  divisor.  Multiply  the  divisor  thus  in- 
creased, into  the  term  last  placed  in  the  root,  and  subtract 
the  product  from  the  dividend. 

Bring  down  two  or  three  additional  terms,  and  proceed  a? 
before. 

Ex.  1.  What  is  the  square  root  of 

a2  +  2a&+62  +2ac-f  2&c-fc2  (a+J+c 

a3,  the  first  subtrahend. 

_  _ 


Into  6=*     2ab+  b  2,  the  2d  subtrahend. 


*       * 
Into     c=  2ac-f26c+c3,  the  3d  subtrahend- 


Here  it  will  be  seen,  that  the  several  subtrahends  are  suc- 
cessively taken  "from  the  given  quantity,  till  it  is  exhausted. 
If  then,  these  subtrahends  are  together  equal  to  the  square  of 
thft  terms  placed  in  the  root,  the  root  is  truly  assigned  by  the 
rule. 

The  first  subtrahend  is  the  square  of  the  first  term  of  the 
root. 

The  second  subtrahend  is  the  product  of  the  second  term 
of  the  root,  into  itself,  and  into  twice  tlie  preceding  term. 

The  third  subtrahend  is  the  product  of  the  third  term  of 
the  root,  into  itself,  und  into  twice  the  sum  of  the  two  pre- 
ceding terms,  &,c. 

That  is,  the  subtrahends  are  equal  to 


and  this  ex.pressian  is  equal  to  the  square  of  the  root. 


EVOLUTION.  245 

For  (a+J)i=as+2a6+4s=a»-4-(2a+5)xi.  (Art.120.) 
And  putting  A  =  a  +  &,  the  square  h  s  =  o  *  -f-  (2a  -f-  J)  x  6. 
And(a+i+c)3=(A+c)a=A*+(2A+c)xcj 
that  is,  restoring  the  values  of  h  and  A2, 

(a+&-fc)3=a3-f-(2a+&)x&+(2a+26+c)xc. 

In  the  same  manner  it  may  be  proved,  that,  if  another 
term  be  added  to  the  root,  the  power  will  be  increased,  by 
frhe  product  of  that  term,  into  itself,  and  into  twice  the  sum 
of  the  preceding  terms. 

The  demonstration  will  be  substantially  the  same,  if  some 
of  the  terms  be  negative. 

2.  What  is  the  square  root  of 


Into  -26  =  — 


2— 46+y)  *      *    2y— 
Into        y  =  2y— Aby+y-. 


3.  What  is  the  square  root  of 

?  Ans.  a3-a 


4.  What  is  the  square  root  of 
o*-r-4aa!i-f462-4aa  -86+4  ?        Ans.  a2  +25-2. 

486.  It  will  frequently  facilitate  the  extraction  of  roots,  to 
consider  the  index  as  composed  of  two  or  more  factors. 

Thus  a^  =  a^  X  *.  (Art.  258.)  And  a?  =  a?  X  *.     That  is, 

The  fourth  root  is  equal  to  the  square  root  of  the  square  ' 
root; 

The  sixth  root  is  equal  to  the  square  root  of  the  cube 
root  ; 

The  eighth  root  is  equal  to  the  square  root  of  the  fourth 
root,  &c. 

To  find  the  sixth  root,  therefore,  we  may  first  extract  the 
cube  root;  and  then  the  square  root  of  this. 

/ 


BE  Ct  10ft   XIX. 


INFINITE  SERIES. 


Aft  497  T^  *s  frecLuently  tne  cas<?>  that,  in  attempting  to 
-*-  extract  the  root  of  a  quantity,  or  to  divide 
one  quantity  by  another,  we  find  it  impossible  to  assign  the 
quotient  or  root  with  exactness.  But,  by  continuing  the  op- 
eration, one  terra  after  another  may  be  added,  so  as  to  bring 
the  result  nearer  and  nearer  to  the  value  required.  See  art. 
454.  When  the  number  of  terms  is  supposed  to  be  extend- 
ed beyond  any  determinate  limits,  the  expression  is  called 
an  infinite  series.  The  quantity,  however,  may  be  finite, 
though  the  number  of  terms  be  unlimited. 

An  infinite  series  may  appear,  at  first  view,  much  less  sim- 
ple, than  the  expression  from  which  it  is  derived.  But  the 
former  is,  frequently,  more  within  the  power  of  calculation 
than  the  latter.  Much  of  the  labour  and  ingenuity  of  math- 
ematicians has,  accordingly,  been  employed  on  the  subject  of 
series.  If  it  were  necessary  to  find  each  of  the  terms  by 
actual  calculation,  the  undertaking  would  be  hopeless.  But 
a  few  of  the  leading  terms  will,  generally,  be  sufficient  to  de- 
termine the  law  of  the  progression. 

488.  A  fraction  may  often  be  expanded  into  an  infinite  se- 
ries, by  dividing  the  numerator  by  the  denominator.  For  the 
value  of  a  fraction  is  equal  to  the  quotient  of  the  numerator 
divided  by  the  denominator.  (Art.  135.)  When  this  quo- 
tient can  not  be  expressed  in  a  limited  number  of  terms,  it 
may  be  represented  by  an  infinite  series. 


1 
divide  1  by  I— a,  according  to  the  rule  io  art.  463. 


Ex.  1 .  To  reduce  the  fraction  .•_  -  to  an  infinile  series, 


INFINITE  SERIES.  247 

o)l       (l-r-cH-aa+a3  &c. 
1-a 


By  continuing  the  operation,  we  obtain  tlje  terms 

l  +  a+a2+a34-#4+a5  -{-a6,  &c.  which  are  sufficient 
to  show  that  the  series,  after  the  first  term,  consists  of  the 
powers  of  a,  rising  regularly  one  above  another. 

That  the  series  may  converge,  that  is,  come  nearer 
and  nearer  to  the  exact  value  of  the  fraction,  it  is  ne- 
cessary that  the  first  term  of  the  divisor  be  greater  than  the 
second.  In  the  example  just  given,  1  must  be  greater  than 
a.  For,  at  each  step  of  the  division,  there  is  a  remainder  ; 
and  the  quotient  is  not  complete,  till  this  is  placed  over  the 
divisor  and  annexed.  Now  the  first  remainder  is  «,  the  sec- 
ond a2,  the  third  a3,  &c.  If  a  then  is  greater  than  1,  the 
remainder  continually  increases  ;  which  shows,  that,  the  far- 
ther the  division  is  carried,  the  greater  is  the  quantity,  either 
positive  or  negative,  which  ought  to  be  added  to  the  quo- 
tient. The  series  is,  therefore,  diverging  instead  of  con- 
verging. 

But  if  a  be  less  than  1,  the  remainders  a,  a9,  a3,  &tc.  will 
continually  decrease.  For  powers  are  raised  by  multiplica- 
tion ;  and  if  the  multiplier  be  less  than  a  unit,  the  product 
will  be  less  than  the  multiplicand.  (Art.  90.)  If  a  be  taken 
equal  to  |,  then  by  art.  223, 

fl*=i,         «3=T»         «*  =  T\»         a'.-,V»fcc. 
and  we  have 


" 


Here,   the  two  first  terms  =1  +•£,  which  is  less  than  2,  by  |  ; 
the  three  first  =  1  +  1,  less  than  2,  by  i  ; 

the  four  first  =•  1  +  -£,  less  than  2,  by 


248  ALGEBRA. 

So  that,  the  farther  the  series  is  carried,  the  nearer  it  ap- 
proaches to  the  value  of  the  given  fraction,  which  is  equal 
to  2. 

1 

2.  If  TV"  be  expanded,  the  series  will  be  the  same  as  that 

from'7 ,  except  that  the  terms  which  consist  of  the  odd 

i  •*"'  a 

powers  of  a  will  be  negative.      fc 

So  that      —  =l-a+«2-a3+o4—  a'+a*  &c. 


h 

3.  Reduce  ~r  to  an  infinite  series. 
bh     o*« 


bh 

a 

bk    b*k 

a  ~"  a* 
b*k 


Here  A  divided  by  a,  gives  —  for  the    first   term  of  the 

quotient.  (Art.  124.)     This  is  multiplied  into  a—  by  and  the 

bh 
product  is  A—  —  ;  (Arts.  159,  153.)  which  subtracted  from 

bh  bh 

h  leaves  —    This  divided  by  a,  gives  -7  (Art.   163.)  for 

the  second  term  of  the  quotient.    If  the  operation  be  con- 
tinued in  the  same  manner,  we  shall  obtain  the  series, 

h     bh     £«A     b*h     b*h 

a+a*+^+~^~+~^~*£C' 
in  which  the  exponents  of  b  and  of  a  increase  regularly  by  I. 


INFINITE  SERIES. 

1+rt 
4.  Reduce   ,__    to  an  infinite  series. 

Ans. 


489.  An  infinite  series  may  be  produced,  by  extracting  th* 
foot  of  a  compound  surd. 

Ex.  "1.    Reduce  Va2+b2   to  an  infinite   series,  by  ex- 
tracting the  square  root,  according  to  the  rule  in  art.  485. 


*1  AL  _!!_ 

+  2a~8a3  ^~16a! 
a2 


b 


f-^-^ 


Here  a,  the  root  of  the  first  terra,  is  taken  for  the  first 
term  of  the  series;  and  the  power  a2  is  subtracted  from  the 
given  quantity.  The  remainder  b3  is  divided  by  2a,  which 

b2 
gives^-,  for  the  second  term  of  the  root.  (Art.  124.)     The 

divisor,  with  this  term  added  to  it,  is  then  multiplied  into  the 

b* 
term,  and  the  product  is  J2-f^-    (Arts.  159,  155.)      This 

b* 
subtracted  from  bz  leaves  —  TTi  which  divided  by  2a  gives 


b* 
—  g~T>  for  the  third  term  of  the  root.  (Art.  163.)  fee. 


2. 
3. 


Hh 


250  ALGEBRA. 

490.  A  binomial  which  has  a  negative  or  fractional  expo- 
nent, may  be  expanded  into  an  infinite  series  by  the  binomi- 
al theorem.  See  arts.  480,  432. 


Ex.  1.  Expand  7—  —  -r-i  =  (rt-j-y)~4  into  an  infinite  series* 
The  terms,  without  the  co-efficients,  are 


-flfrx-6 
The  co-ef.  of  the  2d  term,  is  -4,  of  the  4th  -,  --  ^  -  =-2O, 

-4x-5  ,  -20  x  -7 

of  the  3d,  -  £  --  =  +  10,  of  the  5th,  --  ^  —  =  +  35. 

The  scries  then  is 


Which  is  the  same  (Art.  207.)  as 
1      4y     IQy3      20$  3     35y4 


_\ 


_ 
iJ.    .Expand  (r+y)3  into  an  infinite  series.     See  art.  482; 

Ans.    a*  +  l-a?~*"*y  —  j  a;~  -^2  +  TV^~^3  &c, 
Which  is  the  same  as 
i        11        v2          V* 

"o"     i          *•  *^  i          »  o 

**+—  T~~~+  —  ^~  ***• 
2.rz      8.T2       16z"- 

491.  Though  an  infinite  sei-ies  consists  of  an  unlimited 
number  of  terms,  yet,  in  many  cases,  it  is  not  difficult  to  find 
\vhat  is  called  the  sum  of  the  terms;  that  is,  a  quantity  which 
differs  less,  than  by  any  assignable  quantity,  from  the  value 
of  the  whole.  This  is  also  called  the  limit  of  the  series. 
Thus  the  decimal  0.33333  &c.  may  come  infinitely  near  to 
the  vulgar  fraction  •},  but  can  never  exceed  it,  nor  indeed 
exactly  equal  it.  See  arts.  453,  4.  Therefore  ^  is  the  limit 
of  0.33333  Sec.  that  is,  of  the  series 


3 

Too'fJ 


If  the  number  of  terms  be  supposed  infinitely  great,  the 


INFINITE  SERIES.  251 


•difference  between  their  sura  and  -£>  W*N  be  infinitely 
small. 

492.  The  sum  of  an  infinite  series  whose  terms  decrease 
Ijy  a  common  divisor,  may  be  found,  by  the  rule  for  the  sum 
of  a  series  in  geometrical  progression.  (Art.  442.)  Accor- 

rz  —  a 
ding  to  this,  S=  —  —  y,  that  is,   the   sum  of  the  series  is 

found,  by  multiplying  the  greatest  term  into  the  ratio,  sub- 
tracting the  least  term,  and  dividing  by  the  ratio  less  1.  But, 
in  an  infinite  series  decreasing,  the  least  term  is  infinitely 
small.  It  may  be  neglected,  therefore,  as  of  no  comparative 
value.  (Art.  456.)  The  formula  will  then  become. 

rz—  0  rz 

S=-  or  S==- 

r  —  \  r  —  l 

Ex.  1.  What  is  the  sum  of  the  infinite  series 


Here,  the  first  term  is  T^,  and  the  ratio  is  10. 

TZ          10  X  —  3— 

Then     £=—-=—-  ^  =f  =i,  the  answer. 

T  ~~  JL         1  U  —  ~  JL 

.2.  Whjat  is  the  sum  of  the  infinite  series 


0       rz       2x1 
Ans.   S=__=__^ 


3.  What  is  the  sum  of  the  infinite  series 

c-  Ans-     = 


49?.  There  are  certain  classes  ef  infinite  series,  whose 

may  be  found  by  subtraction. 
By  the  rules  for  the  reduction  and  subtraction  of  fraction?, 

1       1      3-2        1 


a"7"  4  ~"3x4~~3x4 

£     J  __  5-4  __  l_ 
4  ~  5  ~4x5~4x.ri' 


252  ALGEBRA. 

If  then  the  fractions  on  the  right  be  formed  into  a  series, 
they  will  be  equal  to  the  difference  of  two  scries  formed  from 
the  fractions  on  the  left.  This  difference  is  easily  found  : 
for  if  the  first  term  be  taken  away  from  one  of  these  two 
series,  it  will  be  equal  to  the  other. 

Suppose  we  have  to  find  the  sum  of  the  infinite  series 

1111 


From  this,  let  another  be  derived,  by  removing  the  last 
factor  from  each  of  the  denominators ;  and  let  the  sum  of 
the  new  series  be  represented  by  £, 

That  is,  let         S=i  +  $  +  -i4*&c. 
Then  S-=--++    &c. 


And  by  subtraction  -    = 


Here  the  new  series  is  made  one  side  of  an  equation,  and 
directly  under  it,  is  written  the  same  series,  after  the  first 
term  ^  is  taken  away.  If  the  upper  one  is  equal  to  S,  it  is 
.evident  that  the  lower  one  must  be  equal  to  S—  -£.  Then 
subtracting  the  terms  of  one  equation  from  those  of  the 
other,  (Ax.  2.)  we  have  the  sum  of  the  proposed  series 
equal  to  |.  For  S-(S—  '-)  =  «- 


2.  What  is  the  sum  of  the  infinite  series 
11111 


Here  a  new  series  may  be  formed,  as  before,  by  omitting 
the  last  factor  in  each  denominator. 

Let  S=l  +  H-J  +  i  +  |&c. 

Then  ff-fsl+i+i+i+i  be- 


322 
And  by  subtraction      -^-=—-^ 


JL___L   JL  JL  _ L  JL 


INFINITE  SERIES. 

In  repeating  the  new  series,  in  this  case,  it  is  necessary  to 
«mit  the  two  first  terms,  which  are  l  +  ~=f. 

3.  What  is  the  sum  of  the  infinite  series 
1  1  1 


Here  a  new  series  may  he  formed,  by  omitting  the  last 
factor,  and  retaining  the  two  first,  in  each  denominator. 

1111 

Let  S=^.J+^+-6.Q-+j^i0-  &c. 

11111 

Then    S-  -=+-.-++  &c- 


14 
And  b7  subt. 


4.  Wlia.t  is  the  sum  of  the  infinite  series 

1111  1 

&c-       Ans-   * 


*  See  Note  P. 


XX. 


COMPOSITION  AM>  RESOLUTION  or   THE  HIGH;.* 
EQUATIONS. 


AKT  494  "AQUATIONS  of  arty  degree  may  be  produ- 
•^  ced  from  simple  equations,  by  multiplica- 
tion. The  manner  in  which  they  are  compounded  will  be 
best  understood,  by  taking  them  in  that  state  in  which  they 
are  all  brought  on  one  side  by  transposition.  (Art.  178-)  It 
will  also  be  necessary  to  assign,  to  the  same  letter,  different 
values,  in  the  different  simple  equations. 

Suppose,  that  in  one  equation,  a?=2 
And,  that  in  another,  #=3 

By  transposition,  x— 2— 0 

And  a:— 3=0 


Mult,  them  together,  x%  —  5o:-fG=0 

Next,  suppose  #—4=0 

And  multiplying,  a:3  —  9x2  +26r—  24=0 

Again,  suppose  or—  5=0 

And  mult,  as  before  x*  —  Ux3  +  71z*  —  154*+  120=0&c. 
Collecting  together  the  products,  we  have 


That  is, 

The  prod,  of  two  simple  equations,  is  a  quadratic  equation  ; 

of  three  simple  equations,  is  a  cubic  equation  ; 

of  four  simple  equations,   is  a  biquadratic,  or  an 
equation  of  the  fourth  degree,  &c.  (Art.  300.) 

Or,  a  cubic  equation  may  be  considered  as  the  product 


EQUATIONS.  255 

of  a  quadratic  and  a  simple  equation:  a  biquadratic,  as  the 
product  of  two  quadratic ;  or  of  a  cubic  and  a  simple  equa- 
tion, &c. 

495.  In  each  case,  the  exponent  of  the  unknown  quantity, 
in  the  first  term,  is  equal  to  the  degree  of  the  equation ;  and, 
in  the  succeeding  terms,  it  decreases  regularly  by  1,  like  the 
exponent  of  the  leading  quantity  in  the  power  of  a  binomi- 
al. (Art.  463.) 

In  a  quadratic  equation,  the  exponents  are          2,  I. 
In  a  cubic  equation,  3,  2,  1. 

In  a  biquadratic,  4,  3,  2,  1,  &zc. 

496.  The  nwmktr  of  terms,  is  greater  by  1,  than  the  de- 
gree of  the   equation,   or  the   number  of  simple  equations 
irom  which  it  is  produced.      For,  besides  the  terms  which 
contain  the  different  powers  of  the  unknown  quantity,  there 
is  one  which  consists  of  known  quantities  only.      The  equa- 
tion is  here  supposed  to  be  complete*      But  if  there  are,  in 
the  partial  productSj  terms  which  balance  each  other,  these 
may  disappear  in  the  result.   (Art.  110.) 

497.  Each  of  the  values  of  the  unknown  quantity  is  cal- 
led a  root  of  the  equation. 

Thus,  in  the  example  above, 

The  roots  of  the  quadratic  equation  are  3,  2. 

of  the  cubic  equation  4,  3,  2. 

of  the  biquadratic  5,  4,  3,  2. 

The  term  root  is  not  to  be  understood  in  the  same  sense 
here,  as  in  the  preceding  sections.  The  root  of  an  equation 
is  not  a  quantity  which  multiplied  into  itself  will  produce  the 
equation.  It  is  one  of  the  values  of  the  unknown  quantity  ; 
and  when  its  sign  is  changed  by  transposition,  it  is  a  term  in 
one  of  the  binomial  factors  which  enter  into  the  composi- 
sition  of  the  equation  of  which  it  is  a  root. 
•*•  The  value  of  the  unknown  letter  #,  in  the  equation,  is  a 
quantity  which  may  be  substituted  for  #,  without  affecting 
the  equality  of  the  members.  In  the  equations  which  we 
are  now  considering,  each  member  is  equal  to  0;  and  the 
first  is  the  product  of  several  factors.  This  product  will 
continue  to  be  equal  to  0,  as  long  as  any  one  of  it?  factors  is 
0.  (Art.  1 1 2.)  If  then  in  the  equation 

(,T-2)  x  (cc-3)  X  (a -4)  X  (x-5)=Q: 
wo  substitute  2  for  v,  in  the  first  factor,  we  have 
0  x  (s  -  3)  X  (x - 4)  x  (x- 5)  =0, 


256  ALGEBRA. 

So  if  we  substitute  3  for  x,  in  the  second  factor,  or  4  iro 
the  third,  or  5  in  the  fourth,  the  whole  product  will  still  be  0. 
This  will  also  be  the  case,  when  the  product  is  formed  by 
an  actual  multiplication  of  the  several  factors  into  each 
other. 
Thus,  as  x3-9x*  +  26*  -24  =0;  (Art.  494.) 

So  2s  -  9  x  23  +26  x  2-2/1=0, 
And  3s  -9  x  32  +26  x  3-24=0,  &c. 

Either  of  these  values  of  x,  therefore,  will  satisfy  the  con- 
ditions of  the  equation. 

498.  The  number  of  roots,  then,  which  belong  to  any 
equation,  is  equal  to  the  degree  of  the  equation. 

Thus,  a  quadratic  equation  has  two  roots; 
a  cubic  equation,  three; 
a  biquadratic,  four,  &c. 

Some  of  these  roots,  however,  may  be  imaginary.  For 
an  imaginary  expression  may  be  one  of  the  factors  from 
which  the  equation  is  derived. 

499.  The  resolution  of  equations,  which  consists  in  find- 
ing their  roots,  cannot  be  well  understood,  without  bringing 
into  view  a  number  of  principles,  derived  from  the  manner 
in  which  the  equations  are  compounded.      The  laws  by 
which  the  co-efficients  are  governed,  may  be  seen,  from  the 
following  view  of  the  multiplication  of  the  factors 

x  —  a,  x—b,x—c,  x—d, 
each  of  which  is  supposed  equal  to  0. 

The  several  co-efficients  of  the  same  power  of  x,  are  pla- 
ced under  each  other. 

Thus,  —ax—bx  is  written  ~f  >  x'}  and  the  other  co-effi- 
cients, in  the  same  manner. 


The  product,  then 


Of    (x-a}=0 
Into   a—  6=*0 


Is      x2  ~?  £  cr+fl&=0,  a  quadratic  equation, 
This  into  a?—  c=0 


W— «?ic=0.  a 


—a)      +ab 
Is    ,t3  —  b  >  x2  +ac  )>  x— «?'C=0.  a  cubic  equatJor*, 

"Ms  into  r— </=< 


EQUATIONS.  257 

+  ab  ~] 

—abc  ") 

=0,  a  biquadratic. 


500.  By  attending  to  these  equations,  it  will  be  seen  that, 
In  the  first  term  of  each,  the  co-efficient  of  £  is  1 : 
In  the  second  term,  the  co-efficient  is  the  sum  of  all  the 
roots  of  the  equation,  with  contrary  signs.      Thus  the  roots 
of  the  quadratic  equation  are  a  and  b,  and  the  co-efficients,  in 
the  second  term,  are  — a  and  — 6. 

In  the  third  term,  the  co-efficient  of  x  is  the  sum  of  all  the 
products  which  can  be  made,  by  multiplying  together  any 
two  of  the  roots.  Thus,  in  the  cubic  equation,  as  the  roots 
are  a,  6,  and  C,  the  co-efficients,  hi  the  third  term,  are 
«6,  ac,  be. 

In  the  fourth  term,  the  co-efficient  of  x  is  the  sum  of  all 
the  products  which  can  be  made,  by  multiplying  together 
any  three  of  the  roots,  after  their  signs  are  changed.  Thus 
the  roots  of  the  biquadratic  equation  are  a,  6,  c,  and  d,  and 
the  co-efficients  in  the  fourth  term,  are  —  abc,  —abd,  —  acd, 
-bed, 

The  last  term  is  the  product  formed  from  all  the  roots  of 
the  equation,  after  the  signs  are  changed. 

In  the  cubic  equation,  it  is     —ax  —bx  —c^'-abc. 

In  the  biquadratic,         —  ax—  bx  —  ex  —  rf=  +  abed,  &c. 

50L  In  the  preceding  examples,  the  roots  are  all  posi- 
tive. The  signs  are  changed  by  transposition,  and  when  the 
several  factors  are  multiplied  together,  the  terms  in  the  pro- 
duct, as  in  the  power  of  a  residual  quantity,  (Art.  476.)  are 
alternately  positive  and  negative.  But  if  the  roots  are  all 
negative,  they  become  positive  by  transposition,  and  all  the 
terms  in  the  product  must  be  positive.  Thus,  if  the  several 
values  of  *x  are  —a,  —  6,  — c,  —  </,  then 

a;-fa=0,  z-f  b=0,  #+0=0,  x+d=Q', 

and  by  multiplying  these  together,  we  shall  obtain  the  same 
equations  as  before,  except  that  the  signs  of  all  the  terms 
will  be  positive.  In  other  cases,  some  of  the  roots  may  be 
positive,  and  some  of  them  negative. 

li 


ALGEBRA 

502.  As  equations  are  raised,  from  a  lower  degree  to  a 
higher,  by  multiplication,  so  they  may  be  depres&Jh  from  a 
higher  degree  to  a  lower,  by  diiision.  The  product  of 
(x—a)  into  (x  —  Z>)  is  a  quadratic  equation;  thh;  into  (r— c) 
is  a  cubic  equation;  and  tin's  into  (r  — d)  is  a  biquadratic. 
(Art  40  i.)  If  we  reverse  this  process,  and  divide  the  biquad- 
ratic by  (x—d),  the  quotient,  it  is  evident,  will  be  a  cubic 
equation;  and  if  we  divide  this  by  (x—c),  the  quotient  will 
be  quadratic,  &z,c.  The  divisor  is  one  of  'the  factors  from 
•which  the  equation  is  produced,  that  is,  it  is  a  binomial  con- 
sisting of  x  and  one  of  the  roots  with  its  sign  changed. 
When,  therefore,  we  have  found  either  of  the  roots,  we  may 
divide  by  this,  connected  with  the  unknown  quantity,  which 
will  reduce  the  equation  to  the  next  inferiour  degree. 

503i.  Various  methods  have  been  devised  for  the  resolution 
of  the  higher  equations;  but  many  of  them  are  intricate  and 
tedious,  and  other's  are  applicable  to  particular  cases  only. 
The  roots  may  be  found,  however,  with  sufficient  exactness, 
by  successive  approximations.  From  the  laws  of  the  co-effi- 
cients, as  stated  in  art.  500,'a  general  estimate  may  be  form- 
ed of  the  values  of  the  roots.  They  must  be  such,  that, 
when  their  signs  are  changed,  their  product  shall  be  equal  to 
the  last  term  of  the  equation,  and  their  sum  equal  to  the  co- 
efficient of  the  second  term.  A  trial  may  then  be  made,  by 
substituting,  in  the  place  of  the  unsown  letter,  its  supposed 
value.  If  this  proves  to  be  too  small  or  too  large,  it  may  be 
increased  or  diminished,  and  the  trials  repeated,  till  one  is 
found  which  will  nearly  satisfy  the  conditions  of  the  equa- 
tion. After  we  have  discovered  or  assumed  two  approxim- 
ate values,  and  calculated  the  errours  which  result  from  them, 
we  may  obtain  a  more  exact  correction  of  the  root,  by  the 
following 


PHOP  OHTION. 

.  As  the  difference  of  the' errours  t  to  the  difference,  of  the  as- 
sumed numbers : 

So  if  the  least  efrour,.  to  the  correction  required  in  the  cor- 
responding assumed  'nuntoer* 

This  is  founded  on  the  supposition,  that  the  errours  in  the 
results  are  proportioned  to  the  errours  in  the  assumed  numbers. 

:  *Sce  Mutton's -Mathematics. 


EQUATIONS.  :.,r> 

Let  JV  and  n  be  tlie  assumed  numbers  ; 

S  and  s,      the  crrours  of  these  numbers; 
R  and  r,      the  errours  in  the  results. 

Then  by  the  supposition  R:r::S:s 

And  subt.  the  consequents  (Art.3S9.)  R—  r  :  S—s::r:s. 

But  the  difference  of  the  assumed  numbers  is  the  same, 
as  the  difference  of  their  errours.  If,  for  instance,  the  true 
number  is  10,  and  the  assumed  numbers  12  and  15,  the  ef- 
rours  are  2  and  5  ;  and  the  difference  between  2  and  5  is  the 
same,  as  between  12  and  15.  Substituting,  then,  A*—  n  for 
S—s,  we  have  R—  r  :  JV—  «  ::r:  s,  which  is  the  proportion 
stated  above. 

The  term  difference  is  to  be  understood  here,  as  it  is  com- 
monly used  in  algebra,  to  express  the  result  of  subtraction 
according  to  the  general  rule.  (Art.  82.)  In  this  sense,  the 
difference  of  two  numbers  one  of  which  is  positive  and  the 
other  negative,  is  the  same,  as  their  sum  would  be,  if  their 
signs  were  alike.  (Art.  85.) 

The  supposition  which  is  made  the  foundation  of  the  rule 
for  finding  the  true  value  of  the  root  of  an  equation,  is  not 
strictly  correct.  The  errours  in  the  results  are  not  exactly 
proportioned  to  the  errours  in  the  assumed  numbers.  But 
as  a  greater  emnir  in  the  assumed  number,  will  generally  lead 
to  a  greater  errour  in  the  result,  than  a  less  one,  the  rule  will 
answer  the  purpose  of  approximation.  If  the  value  which 
is  first  found,  is  not  sufficiently  correct,  this  may  be  taken  as 
one  of  the  numbers  for  a  second  trial;  and  the  process  may 
be  repeated,  till  the  errour  is  diminished  as  much  as  is  re- 
quired. There  will  generally  be  an  advantage  in  assuming 
two  numbers  whose  difference  is  1,  or  .01,  or  .001,  &,c. 

Ex.  1.  Find  the  value  of  £,  in  the  cubic  equation. 


Here,  as  the  signs  of  the  terms  are  alternately  positive  and 
negative,  the  roots  must  be  all  positive;  (Art.  501.)  their 
product  must  be  10,  and  their  sum  8. 

Let  it  be  supposed  that  one  of  them  is  5'1  or  5'2.  Then, 
substituting  these  numbers  for  a?,  in  the  given  equation,  we 
have, 


260  ALGEBRA. 

Bythelstsuppcs'n,(5'l)3-8x(5-l)2 

By  the  second,        (5'2)s-8  x  (5-2) » +  17  X  (5-2)-10=2-688. 

That  is,  By  the  first  supposition.  By  the  second  supposition. 

The  1st  term,      *3  =     132-651  140-603 

The  2d,          —8*2  =  -208-08  -216-32 

The  3d,           17*  =      '86-7  88-4 

The4th,          -10  =-  10'  -10- 


Sums  or  errours,         +1-271  +  2'658 

Subtracting  one  from  the  other,  1-271 

— —  — 

Their  difference  is  1-417 

Then,  stating  the  proportion, 
1-4 : 0'l  ::  T27 : 0'09,   the  correction  to  be  sub- 
tracted from  the  first  assumed  number  5'1  :  The  remainder 
is  5*01,  which  is  a  near  value  of  x. 

To  correct  this  farther,  assume     x =5-01,  or  5*02 

By  the  first  supposition.  By  the  second  supposition. 
The  1st  term  x*  =     125*751  126-506 

The2d      -8*2  =  —  200*8  —  201'G 

The  3d        17*  =       8547  85-34 

The  4th     —10  =   —10-  -10- 


Errours  +0-121  +  0'246 

0-121 


Difference  0'125 

Then    0-125:0-01  ::  0-121  ;0'01,   the  correction.      This 
subtracted  from  5'01,  leaves  5  for  the  value  of  x',  which  will 
be  found,  on  trial,  to  satisfy  the  conditions  of  the  equation. 
For  53-8x5'  +  17x5-10:=0. 

We  have  thus  obtained  one  of  the  three  roots.      To  find 
the  other  two,  let  the  equation  be  divided  by  #—5,  accor-. 
ding  to  art.  462,  and  it  will  be  depressed  to  the  next  inferi- 
our  degree.  (Art.  502.) 


JJere,  the  equation  becomes  quadratic. 


EQUATIONS.  261 

By  transposition,  x*  -~3#=  —  2 

Completing  the  square,(Art.305.)#2  — 3*  +  £  =  |— 2=£ 
Extract,  and  transp.  (Art.  303.)    *  =  |±^£=-|±£. 

The  first  of  these  values  of  x  is  2  and  the  other  1 . 

We  have  now  found  the  three  roots  of  the  proposed  equa- 
tion. When  their  signs  are  changed,  their  sum  is  —8,  the 
co-efficient  of  the  second  term,  and  their  product  —10  the 
last  term. 

2.  What  are  the  roots  of  the  equation 

x3-8x*  +4*+4S=0?  Ans.  —  2, +4, +  8. 

3.  What  are  the  roots  of  the  equation 

x3-lQx*+G5x-5Q=Q?  Ans.  1,  5,  10* 


*  See  Note 


SECTION    XXI*. 


APPLICATION  or  ALGEBRA  TO  GEOMETRY 

\         m  1     TT  is  often  expedient  to  make  use  of  the  alge- 

AKT.  504.      L  u    •  i  *•      6  r 

braic  notation,  lor  expressing  the  relations  of 

geometrical  quantities,  and  to  throw  the  several  steps  in  a 
demonstration  into  the  form  of  equations.  By  this,  the  na- 
ture of  the  reasoning  is  not  altered.  It  is  only  translated  in- 
to a  different  language.  &giis  are  substituted  for  words,  but 
they  are  intended  to  convey  the  same  meaning.  A  great 
part  of  the  demonstrations  in  Euclid,  really  consist  of  a  se- 
ries of  equations,  though  they  may  sot  be  presented  to  us 
under  the  algebraic  forms.  Thus  the  proposition,  that  the 
siim  of  the  three  angles  of  a  tritngle  is  equal  to  two  right  an- 
gles, (Euc.  32.  1.)  may  be  demonstrated,  either  in  common 
language,  or  by  means  of  the  signs  used  in  algebra. 

Let  the  side  AB,  of  the  triangle  ABC,  (Fig.  1.)  be  con- 
tinned  to  D;  let  the  line  BE  be  parallel  to  AC',  and  let 
GHI  be  a  right  angle. 

The  demonstration,  in  words,  is  as  follows. 

1.  The  angle  EBD  is  equal  to  the  angle  BA  C.  (Euc.  29.1.) 

2.  The  angle  CBE  is  equal  to  the  angle  JlCB. 

3.  Therefore,  the  angle  EBD  added  to  CBE,  that  is,  the  an- 
gle CBD,  is  equal  to  BAG  added  to  JlCB. 

4.  If  to  these  equals,  we  add  the  angle  ABC,  the  angle  . 
CBD  added  to  ABC,  is  equal  to  BJlC  added  to  JlCB 
and  ABC. 

5.  But  CBD  added  to  ABC,  is  equal  to  twice  GHI,  that  is, 
to  two  right  angles.  Euc.  13.  1. 

6.  Therefore,  the  angles  BAC,  and  ACB,  and  ABC,  are 
together  equal  to  twice  GHI,  or  two  right  angles. 

*  This  and  the  follcm  In;;  section  are  to  br.  read  after  the  Elements 
«f  Geometry. 


APPLICATION  TO  GEOMETRY.  263 

Now,  by  substituting  the  sign  -f,  for  the  word  added  or  and, 
and  the  character  =,  for  the  word  equal,  we  shall  have  the 
same  demonstration,  in  the  following  form. 

1.  By  Euclid  29.1.       EBD=BAC 

2.  And  CBE=ACB 

3.  Add.  the  two  equa's,  EBD+  CBE=BAC+  A  CB 

4.  Ad.tftfC'tobothsid's  CBD+ABC=BAG+ACB+ABC 

5.  But,  by  Euc.  13.1,    CBD+ABC=2GIII 
Q,  Mak.the4th<k5th  ecp. 


By  comparing,  one  by  one,  the  steps  of  these  two  demon- 
strations, it  will  be  seen,  that  they  are  precisely  the  same, 
except  that  they  are  differently  expressed.  Ihe  algebraic 
mode  has  often  the  advantage,  not  only  in  being  more  concise 
than  the  other,  but  in  exhibiting  the  order  of  the  quantities 
more  distinctly  to  the  eye.  Thus,  in  the  fourth  and  fifth 
steps  of  the  preceding  example,  as  the  parts  to  be  compar- 
ed are  placed  one  under  the  other,  it  is  seen,  at  once,  what 
must  be  the  new  equation  derived  from  these  two.  This 
regular  arrangement  is  very  important,  when  the  demonstra- 
tion of  a  theorem,  or  the  resolution  of  a  problem,  is  unusu- 
ally complicated.  In  ordinary  language,  the  numerous  rela- 
tions of  the  quantities  require  a  series  of  explanations  io 
make  them  understood;  while,  by  the  algebraic  notation, 
the  whole  may  be  placed  distinctly  before  us,  at  a  single 
view.  The  disposition  of  the  men  on  a  chess-board,  or  the 
situation  of  the  objects  in  a  landscape,  may  be  better  com- 
prehended, by  a  glance  of  the  eye,  than  by  the  most  labour- 
ed description  in  words. 

505.  It  will  be  observed,  that  the  notation  in  the  example 
just  given  differs,  in  one  respect,  from  that  which  is  generally 
used  in  algebra.  Each  quantity  is  represented,  not  by  a 
single  letter,  but  by  several.  In  common  algebra,  when  one 
letter  stands  immediately  before  another,  as  ab,  without  any 
character  between  them,  they  are  to  be  considered  as  multi- 
plied together. 

But,  in  geometry,  AB  is  an  expression  for  a  single  line,  and 
not  for  the  product  of  A  into  B.  Multiplication  is  denoted, 
either  by  a  point,  or  by  the  character  x.  The  product  of 
AB  into'  CD,  is  AB'CD,  or  AB  x  CD. 

500.  There  is  no  impropriety,  however,  in  representing  a 
geometrical  quantity  by  a  single  letter.  We  may  make  & 
stand  for  a  line  or  an  angle,  as  well  as  for  a  number. 


2«4  ALGEBRA. 

If,  in  the  example  above,  we  put  the  angle 


BAC  =&,  CBD=g,  GUI  =/; 

CBE=c, 

the  demonstration  will  stand  thus, 

1.  By  Euc.  29.  1.  a=b 

2.  And  c=d 

3.  Adding  the  two  equations,  a  +  c=g—b+d 

4.  Addiag  A  to  both  sides,  g+h=b-{-  d+h 

5.  By  Euc.  13.  1.  g+k=2l 

G.  Making  the  4th  and  5th  equal,      b+d+  h=2l. 

This  notation  is,  apparently,  more  simple  than  the  other  J 
but  it  deprives  us  of  what  is  of  great  importance  in  geomet- 
rical demonstrations,  a  continual  and  easy  reference  to  the 
figure.  To  distinguish  the  two  methods,  capitals  are  gene- 
rally used,  for  that  which  is  peculiar  to  geometry  ;  and  small 
letters,  for  that  which  is  properly  algebraic.  The  latter  has 
the  advantage,  in  long  and  complicated  processes,  but  the 
other  is  often  to  be  preferred,  on  account  of  the  facility  with 
which  the  figures  are  consulted. 

507.  If  a  line,  whose  length  is  measured  from  a  given  point 
or  line,  be  considered  positive  ;  a  line  proceeding  in  the  op- 
posite direction  is  to  be  considered  negative.  If  AB,  (Fig.2.) 

reckoned  from  DE  on  the  right,  is  positive  ;  A  C  on  the  left 
is  negative. 

A  line  may  be  conceived  to  be  produced  by  the  motion  of 
a  point.  Suppose  a  point  to  move  in  the  direction  of  AB, 
and  to  describe  a  line  varying  in  length  with  the  distance  of 
the  point  from  A.  While  the  point  is  moving  towards  B,  its 
distance  from  A  will  increase.  But  if  it  move  from  B  to- 
wards C,  its  distance  from  A  will  diminish,  till  it  is  reduced 
to  nothing,  and  will  then  increase  on  the  opposite  side.  As 
that  which  increases  the  distance  on  the  riglit,  diminishes  it 
on  the  left,  the  one  is  considered  positive,  and  the  other  neg- 
ative. See  arts.  59,  60. 

Hence,  if  in  the  course  of  a  calculation,  the  algebraic  val- 
ue of  a  line  is  found  to  be  negative  ;  it  must  be  measured  in 
a  direction  to  opposite  that  which,  in  the  same  process,  has 
been  considered  positive.  (Art.  197.) 

508.  In   algebraic  calculations,  there  is  frequent  occasion 
for  multiplication,  division,  involution,  &c.     Bui  how,  it  may 


APPLICATION  TO  GEOMETRY. 

be  asked,  can  geometrical  quantities  be  multiplied  into  each 
other.  One  of  the  factors,  in  multiplication;  is  always  to  be 
considered  as  a  number.  (Art.  91.)  The  operation  consists 
in  repeating  the  multiplicand,  as  many  times  as  there  are 
units  in  the  multiplier.  How  then  can  a  line,  a  surface,  or  a 
solid,  become  a  multiplier? 

To  explain  this,  it  wiil  be  necessary  to  observe,  that  when- 
ever one  geometrical  quantity  is  multiplied  into  another,, 
some  particular  extent  is  to  be  considered  the  unit.  It 
is  immaterial  what  this  extent  is,  provided  it  remain  the 
same,  in  different  parts  of  the  same  calculation.  It  may  be 
an  inch,  a  foot,  a  rod,  or  a  mile.  If  an  inch  is  taken  for  the 
unit,  each  of  the  lines  to  be  multiplied,  is  to  be  considered  as 
made  up  of  so  many  parts,  as  it  contains  inches.  The  mul- 
tiplicand will  then  be  repeated,  as  many  times,  as  there  are 
units  in  the  multiplier.  If,  for  instance,  one  of  the  lines  be 
a  foot  long,  and  the  other,  half  a  foot ;  the  factors  will  be; 
one  12  inches,  and  the  other  6,  and  the  product  will  be  72 
inches.  Though  it  would  be  absurd,  to  say  that  one  line  is 
to  be  repeated,  as  often  as  another  is  long ;  yet  there  is  no 
impropriety  in  saying,  that  one  is  to  be  repeated  as  many 
times,  as  there  are  feet  or  rods  in  the  other*  This,  the  na- 
ture of  a  calculation  often  requires. 

509.  If  the  line  which  is  to  be  the  multiplier,  is  only  a 
part  of  the   length  taken  for  the  unit ;  the  product  is  a  like 
part  of  the  multiplicand.  (Art.  90.)      Thus,  .if  one  of  the 
factors  is  6  inches,  and  the  other  half  aa  inch,  the  product  is 
<3  inches. 

510.  Instead  of  referring  to  the  measures  in  common  use, 
as  inches,  feet,  &c.  it  is  often  convenient  to  fix  upon  one  of 
the  lines  in  a  figure,  as  the  unit  with  which  to  compare  all  the 
others.      When  there  are  a  number  of  lines  drawn  within 
and  about  a  circle,  the  radius  is  commonly  taken  for  the  unit. 
This  is  particularly  the  case  in  trigonometrical  calculations. 

511.  The  observations  which  have  been  made  concerning 
lines,  may  be  applied  to  surfaces  and  solids.     There  may  be 
occasion  to  multiply  the  area  of  a  figure,  by  the  number  of 
inches  in  some  given  line. 

But  here,  another  difficulty  presents  itself.  The  product 
of  two  lines  is  often  spoken  of,  as  being  equal  to  a  surface  ; 
and  the  product  of  a  line  and  a  surface,  as  equal  to  a  solid. 
Thus  the  area  of  a  parallelogram  is  said  to  be  equal  to  the 
product  of  its  base  and  height  j  and  the  solid  contents  of  a 


JUG  ALGEBRA. 

cylinder,  is  >aid  to  be  equal  to  the  product  of  its  length,  into 
I  he  area  of  one  of  its  ends.  But  if  a  line  has  no  breed  (It., 
how  can  the  multiplication,  that  is,  the  repetition,  of  a  lino 
produce  a  surface?  And  if  a  surface  has  no  thickness,  how 
can  a  repetition  of  it  produce  a  solid  ? 

If  a  parallelogram,  represented  on  a  reduced  scale  by 
ABCD,  (Fig.  3.)  be  five  inches  long,  and  three  inches  wide  ; 
the  area  or  surface  is  said  to  be  equal  to  the  product  of  5  in- 
to 3,  that  is,  to  the  number  of  inches  in  AB,  multiplied  by 
the  number  in  BC.  But  the  inches  in  the  lines  *4B  and  BC 
are  linear  inches,  that  is,  inches  in  length  only;  while  those 
which  compose  the  surface  JIC  are  superficial  or  square  in- 
ches, a  different  species  of  magnitude.  How  can  one  of 
these  be  converted  into  the  other  by  multiplication,  a  process 
which  consists  in  repeating  quantities,  without  changing  their 
nature  ? 

512.  In  answering  these  inquiries,  it  must  be  admitted, 
that  measures  of  length  do  not  belong  to  the  same  class  of 
magnitudes  with  superficial  or  solid  measures;  and  that  none 
of  the  steps  of  a  calculation  can',  properly  speaking,  trans- 
form the  one  into  the  other.  But,  though  a  line  can  not  be- 
come a  surface  or  a  soKd,  yet  the  several  measuring  units  in 
common  use  are  so  adapted  to  each  other,  that  squares, 
cubes,  &x.  are  bounded  by  lines  of  the  same  name.  Thus 
the  side  of  a  square  inch,  is  a  linear  inch  ;  that  of  a  square 
rod,  a  linear  rod,  &:c.  The  length  of  a  linear  inch  is  there- 
fore, the  same,  as  the  length  or  breadth  of  a  square  inch. 
If  then,  several  square  inches  are  placed  together,  as  from 

to  R,  (Fig.  3.)  the  number  of  them  in  the  parallelogram 
*H  is  the  same,  as  the  number  of  linear  inches  in  the  side 
and,  if  we  know  the  length  of  this,  we  have  of  course 
the  area  of  the  parallelogram,  which  is  here  supposed,  to  be 
one  inch  wide. 

But.  if  the  breadth  is  several  inches,  the  larger  parallelo- 
gram contains  as  many  smaller  ones,  each  an  inch  wide,  as 
there  aro  inches  in  the  whole  breadth.  Thus,  if  the  paral- 
loloEcratn  *1C  (Fi<r.  3.)  is  5  inches  Ions;,  and  3  inches  broad, 
il  maybe  divided  into  three  such  parallelograms  as  OR.  To 
obtain  then  the  number  of  squares  in  the  large  parallelogram, 
we  have  only  to  multiply  the  number  of  squares  in  one  of 
the  small  parallelograms,  into  the  number  of  such  parallelo- 
grams contained  in  the  whole  figure.  But  the  number  of 
square  inches  in  one  of  the  small  parallelograms,  is  equal  to 
the  number  of  linear  inches  in  the  length  JIB.  And  the 


APPLICATION  TO  GEOMETRY.  26? 

number  of  small  parallelograms,  is  equal  to  the  number  of 
linear  inches  in  the  breadtfTBC.  It  is  therefore  said  con- 
cisely, that  the  area  of  the  parallelogram  is  equal  to  ihc  length 
multiplied  into  the  breadth. 

513.  We  hence  obtain  a  convenient  algebraic  expression 
for  the  area  of  a  right  angled  parallelogram.  If  two  of  the 
sides  perpendicular  to  each  other  arc  JIB  and  />€',  the  ex- 
pression for  the  area  is  JlBxBC;  that  is,  putting  a  for  the 


It  must  be  understood,  however,  that  when  AB  stands  for 
•Aline,  it  contains  only  linear  measuring  units;  but  when  it 
enters  into  the  expression  for  the  area,  it  is  supposed  to  con- 
tain superficial  units  of  the  same  name.  Yet  as,  in  a  given 
length,  the  number  of  one  is  equal  to  that  of  the  other,  they 
may  be  represented  by  the  same  letters,  without  leading  to 
errour  in  calculation. 

514.  The   expression  for  the   area  may  'be  derived,  by  a 
method  more  simple,  but  less  satisfactory  perhaps  to  some, 
from  the  principles  which  have  been  stated  concerning  varia- 
ble quantities,  in  the  13th  section.  Let  a  (Fig.  4.)  represent  a 
square  inch,  foot,  rod,  or  other  measuring  unit;  and  let  b 
and  /  be  t\vo  of  its  sides.      Also,  let  A  be  the  urea  of   any 
right  angled  parallelogram,  B  its  breadth,  and  It  its  length. 
Then  it  is  evident,  that,  if  the  breadth  of  each  were   the 
same,  the  areas  would  be  as  the  lengths  ;  and,  if  the  length 
of  each  were  the  same,  the  areas  would  be  as  the  breadths. 

That  is,         A:a::L:l,  when  the  breadth  is  given, 
And  A  :  a  :  :  B  :  b,  when  the  length  is  given  ; 

Therefore,  (Art.  420.)  «l:a::J3xL:bl,  when  both  vary. 

That  is,  the  area  is  as  the  product  of  the  length  and 
breadth. 

515.  Hence,  in  quoting  the  Elements  of  Euclid,  the  term 
product  is  frequently  substituted  for  rectangle.     And  whatev- 
er is  there  proved   concerning  the  equality  of  certain  rec- 
tangles, may  be  applied  to  the  products  of  the  lines  which 
contain  the  rectangles.* 

516.  The  area  of  an  oblique  parallelogram  is  also  obtain- 
ed, by  multiplying  the  base  into  the  perpendicular  height. 
Thus  the    expression  for  the   area   of  the   paraUelograxa 

*  Soc  Xote  R. 


2C5  ALGEBRA. 

ABNM  (Fig.  5.)  is  MNx  J1D,  or  .,lfi  x  £  C.  For,  by  art. 
.51.3,  ABxBC  is  the  area  of  the  right  angled  parallelogram 
ABCD;  and  by  Euclid  36.1,  parallelograms  upon  equal  bases, 
and  between  the  same  parallels,  are  equal;  that  is,  AB'GD 
is  equal  to  J1BNM. 

>  517.  The  area  of  a.  square  is  obtained,  by  multiplying  one 
of  the  sides  into  itself.  Thus  the  expression  for  the  area 


of  the  square  AC,  (Fig.  G.)  is  JIB  ,lhat  u$, 


For  the  area  is  equal  to  ABxBC.  (Art.  513.) 
But  AB=BC,  therefore,  ABxBC=ABxAB^AB. 

518.  The  area  of  a  triangle  is  equal  to  half  the  product 
jof  the  base  and  height.  Thus  the  area  of  the  triangle  ABG, 
(Fig.  7.)  is  equal  to  half  AB  into  GJ/orits  equal   BC, 
jthat  is, 

a=l  ABxBC. 

For  the  area  of  the  parallelogram  J1BCD  is  ABxBC. 
(Art.  513.)  And,  by  Euc.  41.  1,  if  a  parallelogram  and  a 
triangle  are  upon  the  same  base,  and  between  the  same  par- 
allels, the  triangle  is  half  the  parallelogram. 

519.  Hence,  an  algebraic  expression  may  be  obtained,  for 
the  area  of  any  figure   whatever  which  is  bounded  by  right 
lines.     For  every  such  figure  may  be  divided  into  triangles. 
Thus  the  right-lined  figure 

ABODE  (FigSth.)  is  composed  of  the  triangles 
ABC,ACE,zndECD. 

The  area  of  the  triangle  ABC=\A  C  x  BL, 

That  of  the  triangle  A  CE  =  |«tf  C  X  EH, 

That  of  the  triangle  ECD=$ECxDG. 

The  area  of  the  whole  figure  is,  therefore,  equal  to 


The  explanations,  in  the  preceding  articles,  contain  the 
first  principles  of  the  mensuration  of  superficies.  The  object 
of  introducing  the  subject  in  this  place,  however,  is  not  to 
make  a  practical  application  of  it,  at  present;  but  merely  to 
show  the  grounds  of  the  method  of  representing  geometrical 


quantities  in  algebraic  language. 

520.  Thff  expression  for  the  superficies  has  h 


ere  been  de- 


APPLICATION  TO  GEOMETRY.  260 

rived  from  that  of  a  line  or  lines.  It  is  frequently  necessary 
to  reverse  this  order;  to  find  a  side  of  a  figure,  from  knowing 
its  area. 

If  the  number  of  square  inches  in  the  parallelogram 
JIB  CD  (Fig.  3.)  whose  breadth  RC  is  3  inches,  be  divided 
by  3 ;  the  quotient  will  be  a  parallelogram  AfiEF,  one  inch 
wide,  and  of  the  same  length  with  the  larger  one.  But  the 
length  of  the  small  parallelogram,  is  the  length  of  its  side 
J1R.  The  number  of  square  inches  in  one  is  the  same,  as 
the  number  of  linear  inches  in  the  other.  (Art.  512.)  If 
therefore,  the  area  of  the  large  parallelogram  be  represented 

a 

by  a,  the  side  AB=-^^,  that  is,  the  length  of  a  parallelo- 
gram is  found,  by  dividing  the  area  by  the  breadth. 

521.  If  a  be  put  for  the  area  of  a  square  whose  side  is 


Then  by  art.  517.  a=J!B 

And  extracting  both  sides,  -\fa=JlB 

That  is,  the  side  of  a  square  is  found,  ly  extracting  the 
square  root  of  the  number  of  measuring  units  in  its  area. 

522.  If  JIB  be  the  base  of  a  triangle,  and  B  C  its  perpen- 
dicular height ; 

Then,  by  art.  518, 
And  dividing  by  \BC, 

That  is.  the  base  of  a  triangle  is  found,  by  dividing  the  area 
by  half  the  height. 

523.  As  a  surface  is  expressed,  by  the  product  of  its  length 
and  breadth  ;  the  contents  of  a  solid  may  be  expressed,  by  the 
product  of  its  length,  breadth,  and  depth.     It  is  necessary  to 
bear  in  mind,  that  the  measuring  unit  of  solids  is  a  cube;  and 
that  the  side  of  a  cubic  inch,  is  a  square  inch,  the  side  of  a 
cubic  foot,  a  square  foot,  &c. 

Let  ABCD  (Fig.  3.)  represent  the  base  of  a  parallelopi- 
ped,  5  inches  long,  3  inches  broad,  and  one  inch  deep.  It  is 
evident  there  must  be  as  many  cubic  inches  in  the  solid,  as 
there  arc  square  inches  in  its  base.  And,  as  the  product  of  the 
lines  JIB  and  BC  gives  the  area  of  this  base,  it  gives,  of 
course,  the  contents  of  the  solid.  But  suppose  that  the 
depth  of  the  parallelepiped^  instead  of  being  one  inch,  is 


270  ALGEBRA. 

Jour  inches.  Its  contents  must  be  four  times  as  great.  If, 
then,  the  length  be  AB,  the  breadth  BC,  and  the  depth  CO, 
the  expression  for  the  solid  contents  will  be, 

AJixBCxCO. 

524.  By  means  of  the  algebraic  notation,  a  geometrical 
demonstration  may  often  be  rendered  much  more  simple  and 
•concise,  than  in  ordinary  language.      The  proposition,  (Euc. 
4.  2.)]  that  when  a  straight  line  is  divided  into  two  parts,  the 
square  of  the   whole  line  is  equal  to  the   squares  of  the  two 
parts,  together  with  twice  the  product  of  the  parts,  is  demon- 
strated, by  involving  a  binomial. 

Let  the  side  of  a  square  be  represented  by  s; 

And  let  it  be  divided  into  two  parts,     a  and  b. 

By  the  supposition,  s=a  +  b 

And,  squaring  both  sides,  s 2 _ = a 2  +  2ab -{-u9. 

Or,  changing  the  order  of  the  terms,    sz=a2+i2  +  2ab. 

That  is,  s2  the  square  of  the  whole  line,  is  equal  to  a  a  and 
7;2,  the  squares  of  the  two  parts,  together  with  2ab,  twice  the 
product  of  the  parts. 

525.  The  algebraic  notation   may  also  be  applied,  with 
great  advantage,  to  the  solution  of  geometrical  problems.    In 
doing  this,  it  will  be  necessary,  in  the  first  place,  to  raise  an 
algebraic   equation,  from   the  geometrical   relations  of  the 
quantities  given  and  required;  and  then,  by  the  usual  reduc- 
tions, to  find  the  value  of  the  unknown  quantity  in  this  equa- 
tion.    See  art.  192.. 

Prob.  I.  Given  the  base,  and  the  sum  of  the  hypothen- 
use  and  perpendicular,  of  the  right  angled  triangle,  ABC, 
(Fig.  9.)  to  find  the  perpendicular. 

Let  the  base  AB—l       ") 

The  perpendicular  BC=x 

The  sum  of  hyp.  and  perp.  x+JlC=a        f 
Then  transposing  x,  AC=a—x) 

1 .  By  Euclid  47.  1 ,  'BC2  +  ~AB  -~AC 

2.  That  is,  by  the  notation,     x*+b3=(a-x)*=a3-2ax+ a?*. 

Here  we  have  a  common  algebraic  equation,  containing 
only  one  unknown  quantity.      The  reduction  of  this  equa- 


GEOMETRICAL  PROBLEMS.  271 

tion,  in  the  usual  manner,  will  give  the  value  of  x,  the  side 
required.    ' 

3.  Transp.  and  uniting  terms,  2ax=a2  —  6* 

a3—  b2 

4.  Dividing  by  2a,  x = — ^— — BC. 

The  solution,  in  letters,  will  be  the  same,  for  any  right  an- 
gled triangle  whatever,  and  may  be  expressed  in  a  general 
theorem,  thus;  ' In  a  right  angled  triangle,  the  perpendicu- 
lar is  equal  to  the  square  of  the  sum  of  the  hypothenuse  ancf 
perpendicular,  diminished  by  the  square  of  the  base,  and  di- 
vided by  twice  the  sum  of  the  hypothenuse  and  perpendic- 
ular.' 

It  is  applied  to  particular  eases,  by  substituting  numbers,  for 
the  letters  a  and  b.  Thus,  if  the  base  is  8  feet,  and  the 
sum  of  the  hypothenuse  and  perpendicular  16,  the  expres- 

«2-i3  I6s-82 

sion  — ~ becomes -g— y^— =6,  the  perpendicular;  and 

this  subtracted  from  16,  the  sum  of  the  hypothenuse  and 
perpendicular,  leaves  10,  the  length  of  the  hypothenuse. 

To  prove  that  the  answer  is  correct,  we  have  only  to  ob- 
serve, that  in  conformity  with  Euclid  47.  1,8Z-J-62=102. 

Prob.  2.  Given  the  base,  and  the  difference  of  the  hy- 
pothenuse and  perpendicular,  of  a  right  angled  triangle,  to 
find  the  perpendicular. 

Let  the  base          J5(Fig.  10.)  =b  =20 

The  perpendicular,  BC=x 

The  given  difference",  =d=]Q  ,• 

Then  will  the  hypothenuse  AC=x+d.  J       For  the 

greater  of  two  quantities,  is  equal  to  the  less  ado^d  to  their 

difference.     Then 


1.  By  Euclid  47.  1,  AC  =  AB  +  BC 

2.  That  is,  by  the  notation,  (x+d}*  =b2+x~ 

3.  Expanding  (x+d)*,  (Ait.2'17.)  x2  +2ih+d*  =i3-f.t2 

4.  Transp.  and  uniting  terms,          2,dx=b*  —d* 

b2  —d* 

5.  Dividing  by  2d,  #=— ^—  =  15. 


272  ALGEBftA. 

Prob.  3.  If  the  hypothenuse  of  a  right  angled  triangle  H 
30  feet,  and  the  difference  of  the  other  two  sides  6  feety 
what  is  the  length  of  the  base  ?  Ans.  24  feet. 

Prob.  4.  If  tbe  bypothenusc  of  a  right  angk'd  triangle  is 
50  rods,  and  the  base  is  to  the  perpendicular  as  4  to  3,  what 
is  the  length  of  the  perpendicular?  Ans.  30 

Prob.  5.  Having  the  perimeter  and  the  diagonal  of  a 
parallelogram  ABCD,  (Fig.  11.)  to  find  the  sides. 

I  jet  the  diagonal  AC=h  =  lt" 

The  side  AB=x 

Half  the  perimeter  BC+AB=BC+x=b  =  l<-. 
Then,  by  transposing  x,  £C=b  —  x 

1.  By  Euclid  47.  1,  ~AB  +~BC*  =  ~A~C* 

2.  That  is,  x*  +  (b—x)*  =/ta 

3.  Expanding  (b -x}*,  za+62-26r-K3  =h* 

4.  Transp.  unit,  and  divid.  by  2,  x2  —  6a:=-J/i3  — 15* 

5.  Completing  the  square,          x2-bx+±b^  =^b*+^h2-.*l2 

6.  Extract,  and  transp.  x  =  \btJ \b*-\-^h*-^b*  =8. 

Here  the  side  AB  is  found  ;  and  the  side  B  C  is  equal  to 
£-#=14-8=6. 

Prob.  6.  The  area  of  aright  angled  triangle  ABC  (Fig. 
12.)  being  given,  and  the  sides  of  a  parallelogram  inscribed 
in  it,  to  find  the?side  BC. 

Let  the  given  area  =<*,  DE=BF=b,        ~) 

EB—DF^d,  BC=x. 

Then  by  the  figure,          CF=BC-BF=x^-l. 

1.  By  similar  triangles,          CF:DF::£C :  AB 

2.  That  is,  x—b  :  d::x  \AB 

3.  Mult.  ext.  and  means,      dx=.(x— b}  x  AB 

4.  By  art.  518,  <i=ABx  \B  C=ABx  fz 

5.  Dividing  by  ^r,  ~=AB 

2a  2.a  2(i( 

6.  Subs.in  the  3d,—  for  AB,  dx  =  (x—b)x—=2a  —  - 


7.  Mult,  by  a?,  and  trans,      dx3  —  2az=—  2ab 

2ax         2(tb 

8.  Dividing  by  d,  x2  —  ~J~  =  ~^~j~ 

C2ax      a3      o3     2ab 

9.  Cornplet.  the  square      a2  ~~ 


1  0.  Extract,  and  transp.       x— 


GEOMETRICAL  PROBLEMS.  273 

.  7.  The  three  sides  of  a  right  angled  triangle  ABC, 
(Fig.  13.)  being  given,  to  find  the  segments  made  by  a  per- 
penaicular,  drawn  from  the  right  angle  to  the  hypothenuse. 

The  perpendicular  will  divide  the  original  triangle,  into 
two  right  angled  triangles,  BCD  and  ABD.  (Euc.  8.  6.) 


—  e     , — 2     — 2 


1.  £y  Euc.  47.  1,         ID  +CD=BC 

2.  By  the  figure,  CD=AC-AD 

3.  Squar.  both  sides,     CD=(«tf  C-AD)2 

4.  Therefore, 
3.  Expanding, 
6.  Transposing, 

7.  By  EUC.  47.  i, 

8.  Mak. 

9.  Transposing, 

tO.  DiridingbyS^C,    AD= 

The  unknown  lines,  to  distinguish  them  from  these  which 
are  known,  are  here  expressed  by  Roman  letters. 

Prob.  &.  Having  the  area  of  a  parallelogram  DEFCf 
(Fig.  14.)  inscribed  in  a  given  triangle,  ABC,  to  find  the 
Sides  of  the  parallelogram. 

Draw  CI perpendicular  to  JIB.  By  supposition,  DG  15 
parallel  to  JLB.  '.therefore, 

The  triangle  CHG,  is  similar  <o       CIS  > 
And  CD  if,  to 


Kk 


ALGEBRA. 
Let  CI=d, 


The  given  area     =a. 

1.  By  sim.  trian.  CB:CG::AB: DCf 

2.  And  CB:CG::CI:Cfr 

3.  By  eq.  ratios,  (Art.  384.)     AB :  DG  : :  C/ :  CH 

DGxCI 

4.  Mult.  ext.  and  means, 


5.  By  the  figure,  CI- 

DGxCI 
C.  Substitut.  for  CH,  GI-— 


T.  That  is,  d- 

81.  By  art.  513,  « 

dx* 
9.  That  is,  a*=(lx—-j— 

ID.  Transp.  and  multirby  5,       dx2  —bdx=—ab 

ab 

11.  Dividing  by  J,  oc2  —  bx=—~r 

7,2         7,2 

12.  Completing  the  square,        a 


1-3.  Extract,  and  transp.' 

The  side  DE  is  found,  by  dividing  the  area  by  DG. 

Prob.  9.  Through  a  given  point,  in  a  given  circle,  so  t<* 
draw  a  right  line,  that  its  parts,  between  the  point  and  the 
periphery,  shall  have  a  given  difference. 

In  the  circle  .4.QBR,  (Fig.  15.)  let  P  be  a  given  point,  in 
#ie  diameter  "  n 


Let  AP=a,  PR=x, 

The  given  difference  =d, 
^  will  P 


GEOMETRICAL  PROBLEMS.  27S 


I  .  By  Euc.  35.  3 

2.  That  is,  x  x  (x  +d)  =a 

3.  Or,  afl+dx'^ab 

4.  Completing  the  square,  x  *  -f  dx-\-  ±d2  =^-  d*  +  ah 

5.  Extract,  and  transp.  ~" 


With  a  little  .practice,  the  learner  may  rery  much  abridge 
these  solutions,  and  others  of  a  similar  nature,  by  reducing 
several  steps  to  one.* 


-*  See. Note  S. 


SECTION    XXII, 


EQUATIONS  OF  CURVES, 


_ 
j,         rog     TN  the  preceding  section,  algebra  has  been  ap- 

plied to  geometrical  figures,  bounded  by  right 
lines.  Its  aid  is  required  also,  in  investigating  the  nature  and 
relations  of  curves.  The  advances  which,  in  modern  limes, 
have  been  made  in  this  department  of  geometry,  arc,  in  a 
great  measure,  owing  to  the  method  of  expressing  the  dis- 
tinguishing properties  of  the  different  kinds  of  lines,  in  the 
form  of  equations.  To  understand  the  principles  on  which 
'"inquiries  of  this  sort  are  conducted,  it  is  necessary  to  become 
familiar  with  the  plan  of  notation  which  has  been  generally 
agreed  upon. 

527.  The  posit  ions  of  the  several  points  in  a  curve  Jrauni  on 
G  plane,  are  determined,  by  taking  the  distance  of  each  -from, 
two  right  lines  perpendicular  to  each  other. 

Let  the  lines  AJ?  rind  JIG  (Fig.  16.)  be  perpendicular  to 
each  other.  Also,  let  the  lines  UB,  D'B',  D"B"  be  perpen- 
dicular to  AF;  and  the  lines  CD,  C'D',  C"D",  perpendic- 
ular to  AG.  Then  the  position  of  the  point  D  is 
known,  by  die  length  of  the  lines  BD  and  CD.  In  the 
same  manner,  the  point  D'  is  knoivn,  by  the  lines  B'D'  and 
C'D'  ;  and  the  point  D",  by  the  lines  B  'D"  and  C"D".  The 
two  lines  which  are  thus  drawn,  from  any  point  in  the  curve, 
are,  together,  called  the  co-ordinates  belonging  to  that  point. 
But,  as  there  is  frequent  occasion  to  speak  of  each  of  the 
lines  separately,  one  of  them  for  distinction  sake,  is  called 
an  ordinaie,  and  the  other,  an  abscissa.  Thus  BD  ?s  the  orr 
dinate  of  the  point  D,  and  CD,  or  its  equal  All,  the  nh  • 
scissa  of  the  same  point.  It  is,  generally,  most  convenient 
to  take  the  abscissas  on  the  line  AF,  as  A3  is  equal  to  CD, 
AB  to  CD',  and  AB"  to  C'D".  Euc.  33.  1.  The  lines  AF 
and  AG,  to  which  the  co-ordinates  are  drawn,  are  called  thf 
axes  of  the  co-ordinates. 

528.  If  co-ordinates  could  be  drawn  to  every  point  in  a 
Aurve,  and,  if  the  relations  of  the  sever?,!  abscissas  to  their 


EQUATIONS  OF  CURVES.  277 

corresponding  ordinates  could  be  expressed  by  an  equation  $ 
the  position  of  each  point,  and  consequently,  the  nature  of 
the  curve  would  be  determined.  Many  important  proper- 
ties of  the  figure  might  also  be  discovered,  merely  by  throw- 
ing the  equation  into  different  forms,  by  transposing,  dividing, 
involving,  &c.  But  the  number  of  points  in  a  line  is  unlim- 
ited. It  is  impossible,  therefore,  actually  to  draw  co-ordi- 
nates te  eveiy  one  of  them.  Still  there  is  a  way  in  which 
an,  equation  may  be  obtained,  that  shall  be  applicable  to  all 
the  parts  of  a  curve.  This  is  effected,  by  making  the  equa- 
tion depend  on  some  property,  which  is  common  to  every  pair 
of  co-ordinates.  In  explaining  this,  it  will  be  proper  to  begiri 
/•with  a  straight  line,  instead  of  a  curve. 

Let  AH  (Fig.  17.)  be  a  line  from  which  co-ordinates  arc 
drawn,  on  the  axes  AF  and  AG  perpendicular  to  each  other. 
And  let  the  angle  FAH  be  such,  that  the  abscissa  CD  or  A3 
.shall  be  equal  to  ttvice  the  ordinate  73D. 

The  triangles  ABD,  AB'D',  AB"D",  &c.  are  all  similar. 
,(Euc.  29.  1.)  Therefore, 

.  AB :  BD::AB' :  B'D'-  AB"  :  B"D", 
And  if  AB=2BD,  then  AB'=2B'D',  and  AB"=2B"D\kc. 

That  is,  each  abscissa  is  equal  to  twice  the  corresponding 
jordinate.  But,  instead  of  a  separate  equation  for  each  pair 
of  co-ordinates,  one  will  be  sufficient  for  the  whole.  Let  x 
represent  any  one  of  the  abscissas,  and  y,  the  ordinate  be- 
longing to  the  same  point.  Then, 

This  is.a  general  equation,  expressing  the  ratio  of  the  co- 
ordinates of  the  line  A H  to  each  other.  It  differs  from  a 
common  equation,  in  this,  that  x  and  y,  have  no  determinate 
magnitude.  The  only  condition  which  limits  them  is,  that 
they  shall  be  the  abscissa  and  ordinate  of  the  saine  point. 

If  x-AB,  then  y=BD 

If  x=AB\         y=B'D' 

If  x=AB",        y=B"D",kc. 

From  this  it  is  evident,  that,  if  one  of  the  co-ordinates 
be  taken  of  any  particular  length,  the  other  will  be  given  by 
the  equation.  If,  for  instance,  the  abscissa  x  be  two  inches 
long,  the  ordinate  y,  which  is  half  x,  must  be  one  inch. 


278  ALGEBRA. 

If  A-  =8,  then  t/=4,  If  #=30,  then  y=!S, 

If*=10,        y=5,  If*=100,        y=50,  &r. 

On  the  other  hand,  if  y  —  2,  then  A-  =4,  &c. 

529.  If  the  angle  HAF  be  of  any  different  magnitude,  as 
5ii  Fig.  18,  the  general  equation  will  be  the  same,  except  the 
co-efficient  of  x.  Let  the  ratio  of  y  to  x  be  expressed  by 
4,  that  is,  let  y  :  AT-:  :  a  :  ,1.  'Then  by  converting  this  into  an 
equation,  we  hare 


The  co-efficient  a  will  be  a  whole   number   or  a  frac- 
tion, according  as  y  is  greater  or  less  than  x. 

'530.  To  apply  these  explanations  to  curves,  let  it  be  re- 
«uired  to  find  a  general  equation  of  the  common  parabola. 
(Tig.  19.)  It  is  the  distinguishing  property  of  this  figure,  as 
will  be  shown  under  Conic  Sections,  that  the  abscissas  arc 
jvoportiofied  <to  the  squares  of  their  ordinates.  Let  the  ra- 
tio of  the  square  of  any  one  ordinate  to  its  abscissa,  be  ex- 
pressed by  a.  As  the  ratio  is  the  same,  between  the  square 
of  any  other  ordinate  of  the  parabola  and  its  abscissa,  we 
liavc  universally  y  2  :  #  :,:  #  :  1  ;  and  by  converting  this  into  ap 


This  is  called  the  equation  of  the  curve.  The  important 
advantages  gained  by  this  general  expression,  are  owing  to 
this,  that  the  equation  is  equally  applicable  to  every  point  of 
the  curve.  Any  value  whatever  may  be  assigned  to  the  ab- 
scissa a:,  provided  the  ordinate  y  is  considered  as  belonging  to 
the  same  point.  But,  while  a:  and  y  vary  together,  the 
quantity  a  is  supposed  to  remain  constant. 

By  the,  equation  of  the  parabola  ax  =y3,  and,  extracting 
the  root  of  .both  sides,  (Art.  297.) 

y—Jax.    If  a=2,  then  y=^2x.     And 


If  a?=  4.5 

Ifz=  8.  =AB  y=v/2x]J_  =v/16=4= 

If  x  =  12.5=AB' 

Ifj:=18.  -AB" 


.  When  ordinates  arc  drawn  on  both  sides  of  the  axis 


EQUATIONS  OF  CURVES.  373' 

to  which  they  are  applied  ;  those  on  one  side  will  be  positive* 
while  those  on  the  other  side  will  be  negative.  Thus,  in 
Fig.  19,  if  the  ordinates  on  the  upper  side  of  XFbe  consid- 
ered positive,  those  on  the  under  side  will  be  negative.  ^Art. 
507.)  The  abscissas  also  are  either  positive  or  negative,  ac- 
cording as  they  are  on  one  side  or  the  other  of  the-  point 
fi-orii  which  they  are  measured.  Thus,  in  Fig.  20,  if  the 
abscissas  on  the  right,  AB,  AB',  &cc.  be  considered  positive, 
those  on  the  left,  AC,  AC'",  &:c.  will  be  negative.  And,  iri> 
the  solution  of  a  problem,  if  an  abscissa  or  an  ordinate  i<i 
found  to  be  negative,  it  must  be  set  off,  on  the  side  of  the 
axis  opposite  to  that  on  which  the  values  are  positive. 

532.  In  the  preceding  instances,  the  straight  line  or  curve 
to  which  the  ordinates  and  abscissas  are  applied,  crosses  the 
axis,  in  the  point  where  it  is  intersected  by  the  other  axis, 
Thus  the  curve  (Fig.  19.)  and  the  straight  line  E'D'  (Fig. 
20.)  cross  the  axis  AF,  in  the  point  A,  where  it  is  cut  by  the 
axis  AG.  But  this  is  not  always  the  case.  The  abscissas  on 
the  axis  QF  (Fig.  21.)  may  be  reckoned  from  the  line  GNl 

Let  x  represent  any  one  of  the  abscissas,  MB,  MB',  &cc, 
and  y  the  corresponding  ordinate. 


And  «=the  ratio  of  JSDio  AB,  as  before. 

Then  az=y,  (Art.  529.)  that  is.  z—~ 

a 

But,  by  the  figure,  AB-MB-MA,  i.  e,         z=x-fc 

V 
Making  the  two  equations  equal,  #—  5=~ 

Of 

y 

And  transposing  —  £  x——  -j-iJ. 

ct 

533.  In  investigating  the  properties  of  curves,  it  is  impor-» 
tant  to  be  able  to  distinguish  readily,  the  cases  in  which  the 
abscissas  or  ordinates  prepositive,  from  those  in  which  they  are 
negative  ;  and  to  determine,  under  what  circumstances,  eithe? 
of  the  co-ordinates  vanishes.  An  abscissa  vanishes  at  the  point 
where  the  curve  meets  the  axis  from  which  the  abscissas  are  meas- 
ured. And  an  ordinate  vanishes,  at  the  point  where  the 
curve  meets  the  axis  from  which  the  ordinates  are  measured; 

Thus,  in  Fig.  19,  the  ordinates  are  measured  from  the  lin« 


.280  ALGEBRA 

AF .  Tlie  length  of  each  ordinate  is  the  distance  of  a 
ticular  point  in  the  curve  from  the  line.  As  the  curve  ap- 
proaches tho  axis,  the  ordinal  diminishes,  till  it  become? 
nothing,  at  the  point  of  intersection'.  For,  here,  there  is  no 
distance  between  the  curve  and  the  axis. 

The  abscissas  are  measured  from  the  line  AG.  These 
must  diminish  also,  as  the  curve  approaches  this  line,  and 
become  nothing  at  A. 

534.  From  this  it  is  evident,  that,  when  the  two  axes  meet 
the  curve  at  the  same  point,  the  two  co-ordinates  vanish  to- 
gether.    In  Fig.  19,  the  two  axes  meet  the  curve  at  A,  the 
one  cutting,  and  the  other  touching  it.     But,  in  Fig.  21,  the 
nxis  J\1F  crosses  the  line  JVZ)  at  A;  while  (rJV*  crosses  it  at 
JV.     The  ordinate,  being  the  distance  from  JfaF,  vanishes  at 
A,  where  this  distance  is  nothing.  But  the  abscissa,  being  the 
distance  from  6rJV,-  vanishes  at  JV  or  JW. 

535.  An  abscissa  or  an  ordinate  changes  from  positive  to 
negative,  by  passing  through  the  point  where  it  is  equal  to  0. 
Thus  the  ordinate  y  (Fig.  20.)    diminishes,  as  it  approaches 
the  point  A  ;  here  it  is  nothing,  and,  on  the  other  side  of  A, 
it  becomes  negative,  because  it  is  below  the  axis  CF.   (Art. 
507.)   In  the  same  manner,  the  abscissa,  on  the  right  of  AG, 
diminishes,  as  it  approaches  this  line,  becomes  0  at  A,  and 
then  negative,  on  the  left. 

In  this  case,  the  two  co-ordinates  change  from  positive  to 
negative,  at  the  same  point.  But,  in  Fig.  21,  the  ordinates 
change  from  positive  to  negative  at  A ;  while  the  abscissas- 
continue  positive  to  CUV,  being  still  on  the  right  of  that  line. 
On  the  right  from  A,  the  co-ordinates  are  both  positive  :  be- 
tween A  and  the  line  GN,  the  abscissas  are  positive,  and  the 
<u  Jinatcs  negative  :  and,  on  the  left  of'  GN,  both  are  nega- 
tive. 

536.  The  most   important  applications  of  the  principles 
stated  in  this  section,  will  come  under  consideration,  in  suc- 
ceeding branches  of  the  mathematics,  particularly  in  Flux- 
ions.    A  few  examples,  will  be  here  given,  to  illustrate  th£ 
observations  whkh  have  now  been  made. 

Prob.  1.  To  find  the  equation  of  the  circU. 

In  the  circle  FGM,  (Fig.  22.)  let  the  two  diameters  GJV 
and  FM  be  perpendicular  to  each  other.  From  any  point 
in  the  curve,  draw  the  ordinate  DH  perpendicular  to  .-iF; 
and  AH  will  be  tlip  corresponding  abscissa. 


EQUATIONS  OF  CURVES.  28* 

Let  the  radius  AJ)—r,          AE—x, 


Then,  by  Euc.  47.  1, 

That  is,  y2=r*-x^ 

And  by  evolution,  y=-  ^r*  —  #  a 

In  the  same  nianner,  x=—  Vr2  —  y* 

That  is,  the  abscissa  is  equal  to  the  square  root  of  the  dif- 
ference between  the  square  of  the  radius  and  the  square  of 
the  ordinate. 

If  the  radius  of  the  circle  be  taken  for  a  unit,  (Art.  510.) 
its  square  will  also  be  1,  and  the  two  last  equations  will  be- 
come 

y=±Vl—x2,      and    x-  ±VT^y* 

These  equations  will  be  the  same,  in  whatever  part  of 
the  arc  GDF  the  point  D  is  taken.  For  the  co-ordinates 
will  be  the  legs  of  a  right  angled  triangle,  the  hypothenuse 
of  which  will  be  equal  to  AD,  because  it  is  the  radius  of 
the  circle. 

537.  To  understand  the  application  to  the  other  quarters 
of  the  circle,  it  must  be  observed,  that,  in  each  of  the  equa- 
tions, the  root  is  ambiguous.  The  values  of  y  and  of  x  may 
be  either  positive  or  negative.  This  results  from  the  nature 
of  a  quadratic  equation.  (Art.  297.)  It  corresponds  also 
with  the  situation  of  the  different  parts  of  the  circle,  with 
respect  to  the  two  diameters  FM  and  GN.  In  the  first 
quarter  GF,  the  co-ordinates  are  supposed  to  be  both  posi- 
tive. In  the  second,  GM,  the  ordinates  are  still  positive, 
but  the  abscissas  become  negative.  (Art.  531.)  In  the  third, 
MN,  both  are  negative  :  and  in  the  fourth,  JVF,  the  ordi- 
nates are  negative,  but  the  abscissas  positive.  That  is, 

CFG,  x  is  +3  and  y-\-9 
In  the  qaadra*  ; 


S38,  In  geometry,  lines  are  supposed  to  be  produced  by 
LI 


ALGEBIIA. 

the  motion  of  a  point.  If  the  point  moves  uniformly  in  one 
direction,  it  produces  a  straight  line.  If  it  continually  varies 
its  direction,  it  produces  a  curve.  The  particular  nature  of  the 
curve  depends  on  certain  conditions  by  which  the  motion  is 
regulated.  If,  for  instance,  one  point  moves  in  such  a  man- 
ner, as  to  keep  constantly  at  the  same  distance  from  another 
point  which  is  fixed,  the  fig111'6  described  is  a  circle,  of  which 
the  fixed  point  is  the  centre.  It  is  evident  from  the  preceding 
problem,  that  the  equation  of  this  curve  depends  on  the  man- 
ner of  description.  For  it  is  derived  from  the  property, 
that  different  parts  of  the  periphery  are  equally  distant  from 
the  centre.  In  a  similar  manner,  the  equations  of  other 
curves  may  be  derived  from  the  law  by  which  they  are  de- 
scribed ,  as  will  be  seen  in  the  following  examples? 

Prob.  2.  To  find  the  eqiwition  of  the  curve  called  the 
Cissoid  of  Diodes.  (Fig.  23.) 

The  description,  which  may  be  considered  as  the  defini- 
tion of  the  figure,  is  as  follows. 

In  the  diameter  JlB,  of  the  semi-circle  dtNB,  let  the 
point  R  be  at  the  same  distance  from  jB,  as  P  is  from  A'. 
Draw  RN  perpendicular  to  AB,  to  cut  the  circle  in  N. 
From  .#,  through  JV,  draw  a  straight  line,  extending  if  ne- 
cessary beyond  the  circle.  And  from  Py  raise  a  perpendicular, 
to  cut  this  line  in  M.  The  curve  passes  through  the  point  M. 

By  taking  P  at  different  distances  from  A,  as  in  Fig.  24, 
any  number  of  points  in  the  curve  may  be  determined;  As 
the  line  P,  Jf  moves  towards  .B,  it  becomes  longer  and  longer? 
so  as  to  extend  the  Cissoid  beyond  the  semicircle. 

To  find  the  equation  of  the  .curve,  let  AH  and  AB  be  th# 
axes  of  the  co-ordinates. 


Also,  let  each  of  the  abscissas  AP,  AP'\  AP",  &zc.  =x, 
each  of  the  ordinates  PM,P'M',P'M",  &c.  =  ?/, 
and  the  diameter        JIB  =6, 

Then,  by  the  construction,  PB—AB—AP  =£&  —  x. 

As  PJHand  RN  are  each  perpendicular  to  AB,  the  tri- 
angles  APM  and   ARN  are  similar.  (Euc.  27  and  29.  1.) 

Therefore, 


1  .  By  slm.  triangles,  AP  :  PM:  :  AR  : 

2.  Or,  by  putting  PB  for  its  equal  AR, 


283 


PMxPB 

3.  Mult.  «xt.  and  means,  ~~AP  — 

PM*  x  PB     --  s 

4.  Squaring  .both  sides,          '  —  13^  --  =RN 

AP 

5.  By  Euc.  35.  3,  and  3.  3,    AR  x  RB=RN 

6.  Or,  putting  PJ3  for  its  equal  AR,  and  AP  for  its  equal  RB 

PBxAP=RJV 


PMxPB 

7.  Mak.  4th  and  6th  equal,     PB  x  AP=- 

AP 

8.  Mult.by^anddivid.byP£,  JP=PJVL  xPB 

9.  Or.,  x*=y*x(b-x) 

That  is,  the  cube  of  the  abscissa  is  equal  to  the  square  of 
the  ordinate,  multiplied  by  the  difference  between  the  diam- 
eter of  the  circle,  and  the  abscissa.  The  equation  is  the 
same  for  every  pair  of  co-ordinates. 

Prob.  3.  To  find  the  equation  of  the  Conchoid  of  Nico- 
medes. 

To  describe  the  curve,  let  AB  Fig.  25.  be  a  line  given  in 
position,  and  C  a  point  without  the  line.  About  this  point, 
let  the  line  Ch  revolve.  From  its  intersections  with  AB, 
make  the  distances  EM,  E'M,  E"M",  &c.  each  equal  to 
AD.  The  curve  .will  pass  through  the  points  D,  M,  J\l', 
M",  8tc. 

To  find  its  equation,  let  CD  and  AB  be  the  axes  of  the 
co-ordinates.  Draw  FM  parallel  to  AP,  and  PM  parallel 
to  CF.  Fxom  the  construction,  AD  is  equal  to  EM. 


Let  the  abscissa 

the  ordinate  PM=AF=y, 

the  given  line  CA  =c  a, 

and  AD=EM=b, 

Then  will  <C 


284 


ALGEBRA. 


As  CM  cuts  the  parallels  CD  and  PM,  and  also  the  paraU 
lels  AP  and  FM,  the  triangles  CFM  and  MPE  are        ' 
lar.     Then 


CF:FM:'.PM:P$ 
FMxPM 

T>  E* 


- 

PE  =  -     -2 
CF 

~ 


PE=EM-PM* 
__3   ^_»    FjtfxPM 

~CF 


I ;  By  sim.  triangles, 

2.  Mult.  ext.  and  means, 

3.  Squaring  both  sides, 

4.  By  Euc.  47.  1, 

5.  Mak.  3d  and  4th  equal, 

6.  That  is, 

7.  Mult,  by  (a-f-y)2 

539.  In  these  examples,  the  equation  is  derived  from  the 
.description  of  the  curve.  But  this  order  may  be  reversed. 
If  the  equation  is  given,  the  curve  may  be  described.  For  the 
equation  expresses  the  relation  of  every  abscissa  to  the  cor- 
responding ordinate.  The  curve  is  described,  therefore,  by 
taking  abscissas  of  different  lengths,  and  applying  ordinates  to 
each.  The  line  required,  will  pass  through  the  extremities 
of  these  ordinates. 

Prob.  4.  Te  describe  the  curve  whose  equation  is 

On  the  line 
lengths : 

For  instance,. 4B   =  4'5,then  the  ordinate Z?#=3,(Art.530.} 
AB  =  8-  B'D'     =4, 

AE"  =12-5  JS"D"  =5, 

=13-  J3"'D'"=6, 


^  (Fig.   19.)  take  abscissas  of  different 


Apply  these  several  ordinates  to  their  abscissas,  and  conr 
wect  the  extremities  by  the  line  ADD'D",  &c.  which  will 
i>e  the  curve  required.  The  description  will  be  more  or  less 


EQUATIONS  OF  CURVES.  265 

accurate,  according  to  the  number  of  points  for  which  ordi- 
natcs  are  found. 

540.  If  a  point  is  conceived  to  move  in  such  a  manner,  as 
to  pass  through  the  extremities  of  all  the  ordinates  assigned 
by  an  equation;  the  line  which  it  describes  is  called  the  locus 
of  the  point,  that  is,  the  path  in  which  it  moves,  and  in  which 
it  may  always  be  found.  The  line  is  also  called  the  locus  of 
the  equation  by  which  the  successive  positions  of  the  point 
are  determined.  Thus  the  common  parabola  (Fig.  19.)  is 
called  the  locus  of  the  points  -D,  D',  jD'',  &ic.  or  of  the  cqua- 
.tion  ax=zyz.  (Art.  530.)  The  arc  of  a  circle  is  the  locus 
of  the  equation  x=±  Jr%  —  x2.  (Art.  536.)  To  find  the 
locus  of  an  equation,  therefore,  is  the  same  thing,  as  to  find 
the  straight  line  or  curve  to  which  the  equation  belongs. 

Prob.  5.  To  find  the  locus  of  the  equation 

y 

x=—  ,  or    ax-y, 

in  which,  x  and  y  are  variable  co-ordinates,  while  a  is  a  de- 
terminate quantity. 

If  the  abscissa  x  be  taken  of  different1  lengths,  the  ordi- 
nate  y  must  vary  in  such  a  manner  as  to  preserve  ax=y  ;  or, 
converting  the  equation  into  a  proportion,  y:x::a:l.  There- 
fore, as  a  is  a  determinate  quantity,  the  ratio  of  x  to  y  will 
be  invariable;  that  is,  any  one  abscissa  will  be  to  its  ordinate, 
as  any  other  abscissa  to  its  ordinate.  Let  two  of  the  abscis- 
sas be  JIB  and  JIB',  (Fig.  17.)  and  their  ordinates,  BD  and 
BD';  then, 


The  line  ADD'  is,  therefore,  a  straight  line  :  (Euc.  32.6.) 
and  this  is  the  locus  of  the  equation. 

If  the  proposed  equation  is  x=—  +Z>,  the  additional  term 

Z>,  makes  no  difference  in  the  nature  of  the  locus.  For  the 
only  effect  of  b,  is  to  lengthen  the  abscissas,  so  that  they 
must  not  be  measured  from  A,  but  from  some  other  point,  as 
M,  (Fig.  21.)  The  ratio  of  JIB,  AB',  &,c.  to  BD',  BD'fcc. 
still  remains  the  same.  See  art.  532.  The  locus  of  the 
.equation  is,  therefore,  a  straight  line. 

541.  From  this  it  will  be  easy  to  prove,  that  the  locus  of 
every  equation  in  which  the  co-ordinates  x  and  y  are  in  sepa- 


58«  ALGEBRA. 

rate  terms,  and  do  not  rise  above  the  fast  power  is  a  stralglit 
Jiae.     For  every  such  equation  may  be  brought  to  the  form 
y 

»=  —  ±J.      All  the  terms  may  be  reduced  to  three,  one 
a 

containing  x,  another  y,  and  a  third,  the  aggregate  of  the 
.constant  quantities  which  are  not  co-efficients  of  x  and  y  ;  as 
will  be  seen,  in  the  following  problem. 

Prob.  6.  To  find  the  locus  of  the  equation 
ex—  d+hx—  y+m=n. 

By  transposition,  ex+hx=y+n-~  m-\-d 

y       n—m+d 
Dividing  by  c+h  * 


Here,  the  constant  quantities,  in  each  term,  may  be  repre- 
sented by  a  single  letter.(Art.321.)  If,  then,  we  make  c-fA=a, 

ft_?#-J-{/  y 

and  ---—  r-=&;  the  equation  will  become  #=  —  4&,  whose 

C  *T*  ft  %M 

locus,  by  the  last  article,  is  a  straight  h'ne. 

542.  But  if  the  ordinates  are  as  the  squares,  cubes,  or 
higher  powers  of  the  abscissas,  the  locus  of  the  equation  in- 
stead of  being  a  straight  line,  is  a  curve.      Far  the  ordinates 
applied  to  a  straight  line,  have  the  same  ratio  to  each  other 
which  their  abscissas  have.      But  quantities  have  riot  the 
same  ratio  to  each  other,  which  their  squares,  cubes,  or  high- 
er powers  have.  (Art.  354.)      Thus,  if  #2=y,  the  ordinales 
•vvHl  increase  more  rapidly  than  the  abscissas.  If  the  abscissas 
be  taken,  1,  2,  3,  4,  &,c.  the  ordinates  will  be  equal  to  then- 
squares,  1,  4,  9,  16,  •&&. 

543.  As  an  unlimited  variety  of  equations  may  be  produ- 
ced, by  different  combinations  and  powers  of  the  co-ordi- 
nates, and  as  each  of  these  has  its  appropriate  locus;  it  is  ev- 
ident that  the  forms  of  curves  must  be  innumerable.      They 
may,  however,  be  reduced  to  classes.     The  modern  mode  of 
classing  them,  is  from  the  degree  of  their  equations.      The 
different  orders  of  lines  are  -distinguished,  by  the  greatest  index, 
cr  sum  of  the  indices  of  the  co-ordinates,  in  any  term  of  the 
equation. 

Thus  the  equation  a-x=y  belongs  to  a  iine  of  the  first  or- 
dei\  because  the  index  of  each  of  the  co-ordinates  is  1  .  But 
this  order  includes  no  curves.  For,  by  art.  541,  the  locus  of 
every  such  equation  is  a  straight  line.. 


EQUATIONS  OF  CURVES.  237 

The  equation  cxs  —axy^y'1,  belongs  to  the  second  order 
of  lines,  or  the  first  kind  of  cutves,  because  the  greatest  in- 
dex is  2.  The  equation  ay+xy—lx  also  belongs  to  the  sec- 
ond order.  For,  although  there  is  here  no»  index  greater 
than  1,  yet  the  sum  of  the  indices  of  x  and  y,  in  the  second 
term,  is  2. 

The  equation- y3  —3axy=bxz  belongs  to  the  third  order  of 
lines,  or  the  second  kind  of  curves,  because  the  greatest  in- 
dex of  y  is  3. 

544.  In  curves  of  the  higher  orders,  the  ordinate  belong- 
ing to  any  given  abscissa  may  have  different  values,  and  may 
therefore  meet  the  curve  in  several  points.  For  the  length 
of  the  ordinate  is  determined  by  the  equation  of  the  curve,, 
and  if  the  equation  is  above  the  first  degree,  if  may  have 
two  or  more  roots,  (Art.  498.)  and  may,  therefore,  give  dif- 
ferent values  to  the  ordinate. 

An  equation  of  the  first  degree  has  but  one  'root;  and  a 
line  of  the  first  order,  can  be  intersected  by  an  ordinate,  irr 
one  point  only.  Thus  the  equation  of  the  line  AH  (Fig.  17.) 
is  ax==y,  in  which  it  is  evident  y  has  but  one  value,  while  x- 
remains  the  same.  If  the  abscissa  x  be  taken  equal  to  AB, 
the  ordinate  y  will  be  BD,  which  can  meet  the  line  All  i&> 
D  only. 

But  the  equation  of  the  parabola,  y*—ax,  (Art.  530.)  has 
two  roots.  For,  by  extracting  both  sides,  y—±^/ax.  (Art. 
297.)  It  is  true  that,  in  this  case,  the  two  values  of  y  are 
equal.  But  one  is  positive,  and  the  other  negative.  This 
shows  that  the  ordinate  may  extend  both  ways  from  the  end 
of  the  abscissa,  and  may  meet  the  opposite  branches  of  the 
curve.  Thus  the  ordinate  of  the  abscissa  JIB  (Fig.  19.^ 
may  be  either  BD  above  the  abscissa,  or  Bd  below  it. 

A  cubic  equation  has  three  roots;  and  an  ordinate  of  the 
curve  belonging  to  this  equation,  may  have  three  different 
values,  and  may  meet  the  curve  in  three  different  points. 
Thus  the  ordinate  of  the  abscissa  AB  (Fig.  26.)  may  be  BD-, 
or  BD',  or  Bd. 

545.  When  the  curve  meets  the  axis  on  which  the  ab- 
scissas are  measured,  the  ordinate,  after  becoming  less  and 
less,  is  reduced  to  nothing.  (Art  533.)  But,  in  some  eases,. 
a  curve  may  continually  approach  a  line,  without  ever  meet- 
ing it.  Let  the  distances  AB,  EB' ,  B'B',  &:c.  on  the  line 
AF,  (Fig.  27.)  be  equal;  and  let  the  curve  DDD,  &c.  be 
«f  such  a  nature,  that,  of  the  several  ordinals  at  the  points 


2&8  ALGEb'KA. 

By  B\  B",  &e.  ear.h  succeeding  one  shall  be  half  the  prece- 
ding, that  is,  B  D'  half  BD,  B"D"  half  BD',  &c.  It  is  evi- 
dent, that,  however  far  the  straight  line  be  carried,  the  curve 
will  be  coming  nearer  and  nearer  to  it.  and  yet  will  never 
quite  reach  it.  Jl  line  ivhich  thus  continually  approaches  a 
curve,  icithoiU  ever  meeting  it,  is  called  an  ASSYMPTOTE  of  the 
curve.  The  axis  slF  is  here  the  assymptote  of  the  curve 
DD'D",  &c.  As  the  abscissa  increases,  the  ordinate  dimin- 
ishes, so  that,  when  the  abscissa  is  mathematically  infinite, 
(Art.  447.)  the  ordinate  becomes  an  infinitesimal,  and  may 
l»e  expressed  by  0.  (Art.  455.)* 

*  See  Note  T. 


NOTE  A.    Page  40. 


~j"T  is  common;  to  define  multiplication,  by  saying  that  'it  is 
^  finding  a  product  which  has  the  same  ratio  to  the  multi- 
plicand, that  the  multiplier  has  to  a  unit.'  This  is  strictly  and 
universally  true.  But  the  objection  to  it,  as  a  definition,  is, 
that  the  idea  of  ratio,  as  the  term  is  understood  in  arithmetic 
and  algebra,  seems  to  imply  a  previous  knowledge  of  multi- 
plication, as  well  as  of  division.  In  this  work,  at  least,  geo- 
metrical ratio  is  made  to  depend  on  division,  and  division,  on 
multiplication.  Ratio,  therefore,  could  not  be  properly  in- 
troduced into  the  definition  of  multiplication. 

It  is  thought,  by  some,  to  be  absurd  to  speak  of  a  unit  as 
consisting  of  parts.  But,  whatever  may  be  true  with  respect 
to  number  in  the  abstract,  there  is  certainly  no  absurdity  in 
considering  an  integer,  of  one  denomination,  as  made  up  of 
parts  of  a  different  denomination.  One  rod  may  contain: 
several  feet ;  one  foot,  several  inches,  &tc.  And  in  multipli- 
cation, we  may  be  required  to  repeat  the  whole,  or  a  part 
of  the  multiplicand,  as  many  limes,  as  there  are  inches  in 
a  foot,  or  part  of  a  foot. 


NOTE  B.     p.  97. 

As  the  direct  powers  of  an  integral  quantity  have  positive 
indices,  while  the  reciprocal  powers  have  negative  indices  j 
it  is  common  to  call  the  former  positive  powers,  and  the  lat- 
ter negative  poivers.  But  this  language  is  ambiguous,  and 
may  lead  to  mistake.  For  the  same  terms  are  applied  to 
powers  with  positive  and  negative  signs  prefixed.  Thus 
+  8a4  is  called  a  positive  power;  while  — 8a4  is  called  a 
negative  one.  It  may  occasion  perplexity,  to  speak  of  the 
latter  as  being  both  positive  and  negative  at  the  same  time ; 
positive,  because  it  has  a  positive  index,  and  negative,,  because 

Mm 


200  ALGEBRA. 

it  has  a  negative  co-efficient.  This  ambiguity  may  be  av&id- 
cd,  by  usiiig  the  terms  direct  and  reciprocal;  meaning,  by 
the  former,  powers  with  positive  exponents,  and,  by  the  lat- 
ter, powers  with  negative  exponents. 

NOTE  C.     p.  151. 

Every  affected  quadratic  equation  may  be  reduced  to  en*1 
of  the  three  following  forms. 


3.  x*— ax=—  6) 
These,  when  they  arc  resolved  become 


In  the  two  first  of  these  forms,  the  roots  arc  never  ima- 
ginary. For  the  terms  under  the  radical  sign  are  both  posi- 
tive. But,  in  the  third  form,  whenever  6  is  greater  than 
ias,  the  expression  £aa—  b  is  negative,  and  therefore  its 
root  is  impossible^ 


NOTED.    p.  173. 

This  definition  of  compound  ratio  is  more  comprehen- 
sive than  the  one  which  is  given  in  Euclid.  That  is  included 
in  this,  but  is  limited  to  a  particular  case,  which  is  stated  in 
art.  353.  It  may  answer  the  purposes  of  geometry,  but  is 
not  sufficiently  general  for  algebra. 


NOTE  E.     p;  180i 

It  is  not  denied,  that  very  respectable  writers  use  these 
terms  indiscriminately.  But  it  appears  to  be  without  any 
necessity.  The  ratio  of  6  to  2  is  3;  There  is  certainly  a 
difference  between  ttcice  this  ratio,  and  the  square  of  it,  that 
is,  between  twice  three,  and  the  square  of  three.  All  are 
agreed  to  call  the  latter  a  duplicate,  ratio.  What  occasion  if 


NOTES.  291 

there,  then,  to  apply  to  it  the  term  double  also?  This  is 
wanted,  to  distinguish  the  other  ratio.  And  if  it  is  confined 
to  that,  it  is  used  according  to  the  common  acceptation  of 
ike  word,  in  familiar  language. 


NOTE  F.     p.  155. 

The  definition  here  given  is  meant  to  be  applicable  to 
quantities  of  every  description.  The  subject  of  proportion, 
as  it  is  treated  of  in  Euclid,  is  embarrassed  by  the  means 
which  are  taken  to  provide  for  the  ease  of  incommensurable 
quantities.  But  this  difficulty  is  avoided  by  the  algebraic  no- 
tation, which  may  represent  the  ratio  even  of  incommensu- 
rables. 

1 

Thus  the  ratio  of  1  to  v/2  is  — :=-  • 

•/z 

It  is  impossible  indeed,  to  express,  in  rational  numbers, 
-ihe  square  root  of  2,  or:the  ratio  which  it  bears  to  1.  But 
this  is  not  necessary,  for  the  purpose  of  showing  its  equality 
with  another  ratio. 

The  product  4x2=3.] 
And,  as-equal  quantities  have  equal  roots, 

2  X  v/2 =/8,  therefore,  2 :  ^8  ;*  1 :  V2. 

Here  the  raijo  of  2  to  V8»  *s  proved  to  be  the  same,  as 
that  of  1  to  Y/.2 ;  although  we  are  unable  to  find  the  exact 
value  either  of  ^8  or  ^/2. 

It  is  impossible  to  determine,  with  perfect  accuracy,  the 
ratio  which  the  side  of  a  square  has  to  its  diagonal.  Yet  it 
is  easy  to  prove,  that  the  side  of  one  square  has  the  same  ra- 
tio to  its  diagonal,  which  the  side  of  any  other  square  has  to 
its  diagonal.  When  incommensurable  quantities  are  once 
reduced  to  a  proportion,  they  are  subject  to  the  same  laws  as 
other  proportionals.  Throughout  the  section  on  pro- 
portion, the  demonstrations  do  not  imply  that  we  know  the 
value  of  the  terms,  or  their  ratios;  but  only  that  one  of  the 
ratios  is  equal  to  the  other. 


292  ALGEBRA. 

NOTE  G.     p.  190. 

The  inversion  pf  the  means  can  be  made,  with  strict  pro- 
priety, in  those  cases  only  in  which  all  the  terms  are  quantir 
lies  of  the  same  kind.  For,  if  the  two  last  he  different 
from  the  two  first,  the  antecedent  of  each  couplet,  after  the 
inversion,  will  be  different  from  the  consequent,  and  there- 
fore, there  can  be  no  ratio  between  them.  (Art.  355.) 

This  distinction,  however,  is  of  little  importance  in  prac- 
tice. For,  when  the  several  quantities  are  expressed  in 
numbers,  there  will  always  be  a  ratio  between  the  numbers. 
And  when  two  of  them  are  to  be  multiplied  together,  it  is 
immaterial  which  is  the  multiplier,  and  which  the  multipli- 
cand. Thus,  in  the  Rule  of  Three  in  arithmetic,  a  change 
jn  the  order  of  the  two  middle  terms  will  make  no  difference 
in  the  result. 


NOTE  H,     p.  197. 

The  terms  composition  and  division  are  derived  from  ge- 
ometry, and  are  introduced  here,  because  they  are  generally 
.used  by  writers  on  proportion.  But  they  are  calculated  rath- 
er to  perplex,  than  to  assist,  the  learner.  The  objection  to 
the  word  composition  is,  that  its  meaning  is  liable  to  be  mis- 
taken for  the  composition  or  compounding  of  ratios.  (Art. 
390.)  The  two  cases  are  entirely  different,  and  ought  to  be 
carefully  distinguished.  In  one,  the  terms  are  added,  in  the 
other,  they  are  "multiplied  together.  The  word  compound 
has  a  similar  ambiguity  in  other  parts  of  the  mathematics. 
The  expression  a  +  b,  in  which  a  is  added  to  b,  is  called  a 
compound  quantity.  The  fraction  -£  of  |,  or  £  x  f ,  in  which 
^  is  multiplied  into  |-,  is  called  a  compound  fraction. 

The  terra  division,  as  it  is  used  here,  is  also  exceptionable. 
The  alteration  to  which  it  is  applied,  is  effected  by  subtrac- 
tion, and  has  nothing  of  the  nature  of  what  is  called  division 
in  arithmetic  and  algebra.  But  there  is  another  case,  (Art. 
392.)  totally  distinct  from  this,  in  which  the  change  in  the 
surms  of  the  proportion  is  actually  produced  by  division. 


NOTE  I.     p.  203. 
The  principles  stated  in  this  section,  are  not  only  cxpres- 


NOTES.  293 

sod  in  different  language,  from  the  corresponding  propositions 
in  Euclid,  but  are,xin  several  instances,  more  general.  Thus 
the  first  proposition  in  the  fifth  book  of  the  Elements,  is  con- 
lined  to  equimultiples.  But  the  article  referred  to,  as  con- 
taining this  proposition,  is  applicable  to  all  cases  of  equal 
ratios,  whether  the  antecedents  are  multiples  of  the  conse- 
quents or  not. 

NOTE  K.     p.  217. 

The  solution  of  one  of  the  cases  is  omitted  in  the  text, 
because  it  is  performed  by  logarithms,  with  which  the  learn- 
er is  supposed  not  to  be  acquainted,  in  this  part  of  the  course. 
When  the  first  term,  the  last  term,  and  the  ratio,  are  given, 
the  number  of  terms  may  be  found  by  the  formula 

rz 

.-^ 

n~log.  r' 


NOTE  L.     p.  221. 

When  it  is  said  that  a  mathematical  quantity  may  be  sup- 
posed to  be  increased  beyond  any  determinate  limits,  it  is 
not  intended  that  a  quantity  can  be  specified  so  great,  that 
no  limits  greater  than  this  can  be  assigned.  The  quantity 
and  the  limits  may  be  alternately  extended  one  beyond  the 
other.  If  a  line  be  conceived  to  reach  to  tRe  most  distant 
point  in  the  visible  heavens,  a  limit  may  be  mentioned  be- 
yond this.  The  line  may  then  be  supposed  to  be  extended 
farther  than  this  limit.  Another  point  may  be  specified  still 
farther  on,  and  yet  the  line  may  be  conceived  to  be  carried 
beyond  it. 


NOTE  M.     p.  223. 

The  apparent  contradictions  respecting  infinity,  are  ovrhiaj 
to  the  ambiguity  of  the  term.  It  is  often  thought  that  the 
proposition,  that  quantity  is  infinitely  divisible,  involves  an 
absurdity.  If  it  can  be  proved  that  a  line  an  inch  long  can 
be  divided  into  an  infinite  number  of  parts,  it  can,  by  the 
made  cf  reasoning,  be  proved,  that  a  line  two  inchc? 


294  ALGEBRA. 

iong  may  be  first  divided  in  the  middle,  and  then  tack  of  tine 
sections  be  divided  into  an  infinite  number  of  parts.  In  this 
way,  we  shall  obtain  one  infinite  twice  as  great  as  another. 

If  by  infinity,  here,  is  meant  that  which  is  beyond  any  as^ 
;ignable  limits,  one  of  these  infinites  may  be  supposed  great- 
er than  the  other,  without  any  absurdity.  But  if  it  be  meant 
that  the  number  of  divisions  is  so  great  that  it  can  not  be 
increased,  we  do  not  prove  this,  concerning  either  of  the 
lines.  We  make  out,  therefore,  no  contradiction.  The  ap- 
parent absurdity  arises  from  shifting  the  meaning  of  the 
terms.  We  demonstrate  that  a  quantity  is,  in  one  sense,  infir 
nite ;  and  then  infer  that  it  is  infinite,  in  a  sense  widely  dif- 
ferent. 

NOTE  N.    p.  227. 

Strictly  speaking,  the  inquiry  io  be  made  is,  how  often  the 
•whole  divisor  is  contained  in  as  mauy  terms  of  the  dividend. 
But  it  is  easier  to  divide  by  a  part  only  of  the  divisor ;  and 
this  will  lead  to  no  errour  in  Lbe  resist,  as  the  whole  divisor 
is  multipliedj  in  obtaining  the  several  subtrahends. 

•i    -rV-  j'iii-iifEi'i 

NOTE  O.     p.  235. 

The  demonstration  of  tliis  proposition,  particularly  in  its 
application  to  fractional  indices,  could  not  be  introduced, 
with  advantage,  in  this  part  of  ihe  course.  It  does  not  ap- 
pear that  Newton  himself  demonstrated  his  theorem,  except 
by  induction.  And  though  various  demonstrations  have  since 
been  given ;  yet  they  .are  generally  founded  upon  principles 
and  methods  of  investigation  not  contained  in  this  introduc- 
tion, such  as  the  laws  of  combinations,  fluxions,  and  figu- 
rate  numbers. 

Those  who  wish  to  examine  the  inquiries  on  this  subject, 
may  consult  Simpson's  Algebra,  Section  15,  Euler  s  Algehra, 
Section  ii.  Chap.  11,  Vince's  Fluxions,  Art.  99,  Lacroix's 
Algebra,  Art.  138,  &:c.  Do.  Comp.  Art.  71,  Rees'  Cyclope- 
dia, Manning's  Algebra,  and  the  London  Phil.  Trans.  Vol. 
xxxv.  p.  298. 

NOTE  P.     p.  253. 
The  very  limited  extent  of  this  work  would  admit  of  no» 


NOTES. 

thing  naol-e,  than  a  bare  specimen  of  the  Summation  of  Series. 
For  information  oh  this  subject,  the  learner  is  referred  to 
Kind-son's  Method  of  Increments,  Sterling's  Summation  of 
Merits,  Waring's  Fluxions,  Maclaurin's  Fluxions,  Art  828, 
&ic.  Wood's  Algebra,  Art.  410,  Lacroix's  Comp.  Alg.  Art. 
81,  fcc.  Euler's  Anal.  Intin.  C.  x-in.  Simpson's  Essays  and 
Dissertations,  De  Moivre's  Miss.  Analyt.  p.  72,  and  the  Lon- 
don Philosophical  Transactions. 


NOTE  Q.     p.  261. 

To  those  who  have  made  any  considerable  progress  in  tlwi* 
mathematics,  this  section  will  doubtless  appear  very  defec- 
tive. But  it  was  impossible  to  do  juctice  to  the  subject, 
without  occupying  more  room,  than  could  be  allotted  to  it 
here.  In  going  through  an  elementary  course  of  mathemat- 
ics and  natural  philosophy,  the  student  wHl  rarely  have  occa- 
sion to  solve  an  equation  above  the  second  degree. 

Those  who  wish  to  examine  particularly  the  different  meth- 
ods of  solution,  will  find  them,  irr  Newton's  Universal  Arith- 
metic, Maclaurin's  Alg.  Part  n,  Euler's  Alg.  Part  1,  Sec,  4, 
Waring's  Algebra,  Do.  Medit.  Algeb.  Wallis'  Algebra,  Simp- 
son's Alg.  Sec.  12,  Fenn-'s  Alg.  Ch.  3  and  4.  Saunderson's 
Alg.  Book  x,  Simpson's  Essays  and  Dissertations,  Journal  De 
Physique,  Mar.  1807,  and  Philosophical  Transactions. 


NOTE  R.    p.  267, 

It  will  be  thowght,  perliaps,  that  it  was  unnecessary  to  be 
so  particular,  in'  obtaining  the  expression  for  the  area  of  a 
parallelogram,  for  the  use  of  those  who  read'  Playfair's  edi- 
tion of  Euclid,  in  which  "  JlD.DC  is  put  for  the  rectangle 
contained  by  «/2/^and  DC."  It  is  to  be  observed,  however, 
that  he  introduces  this,  merely  as  an  article  of  notation. 
(Book  ii.  Def.  1.)  And  though  a  point  interposed  between 
the  letters,  is,  m  algebra,  a  sign  of  multiplication;  yet  he 
does  not  here  undertake  to  shew  how  the  sides  of  a  paraDel- 
ogvam  may  be  multiplied  together.  In  the  first  book  of  the 
Supplement,  he  has  indeed  demonstrated,  that  "equiangukr 
parallelograms  are  to  one  another,  as  the  products  of  the 
numbers  proportional  to  their  sides."  But  he  has  not  giv- 
^n  to  the  expressions,  the  forms  most  convenient  for  the 


296  ALGEBRA. 

succeeding  parts  of  this  work.  In  making  the  transition 
from  pure  geometry  to  algebraic  solutions  and  demonstra- 
tions, it  is  important  to  have  it  clearly  seen,  that  the  geomet- 
rical principles  are  not  altered  j  but  are  only  expressed  in  a 
different  language. 


NOTE  S.     p.  275. 

This  section  comprises  very  little  of  what  is  commonly  un- 
derstood by  the  application  of  algebra  to  geometry.  The 
principal  object  has  been,  to  prepare  the  way  for  the  other 
parts  of  the  course,  by  stating  the  grounds  of  the  algebraic 
notation  of  geometrical  quantities,  and  rendering  it  familiar 
by  a  few  examples. 

On  the  construction  and  solution  of  problems,  see  New- 
ton's Arithmetic,  Simpson's  Alg.  Sec.  18  and  appendix,  La- 
rroix's  App.  Alg.  Geom.  Saunderson's  Alg.  Book  xm,  Ana- 
lyt.  Instit.  of  Maria  Agncsi,  Book  1,  Sec.  2,  and  Emerson's- 
Alg.  Book  ii.  Sec.  6. 

NOTE  T.    p.  288, 

On  the  equations  of  curves,  the  geometrical  construction 
of  equations,  the  finding  of  loci,  &c.  see  Maclaurin's  Alg, 
Part  in,  and  appendix,  Newton's  Arith.  Emerson's  Alg.  Book 
ii.  Sec.  9,  Do.  Prop,  of  Curves,  Euler's  Anal.  Infill.  War- 
ing's  Prop.  Alg.  and  Mansfield's  Essays, 

Among  tbe  subjects  which,  for  want  of  room,  are  entirely 
omitted  in  this  introduction,  one  of  the  most  interesting  i's 
the  indeterminate  analysis.  No  part  of  algebra,  perhaps,  is 
better  calculated  to  exercise  the  powers  of  invention.  But 
other  branches  of  the  mathematics  are  so  little  dependent  on 
this,  that  it  is  not  absolutely  necessary  to  give  it  a  place  in 
an  elementary  course. 

See.  on  this  subject,  Euler's  Algebra,  Vol.  u,  with  La- 
grange's  additions,  Saunderson's  Alg.  Book  vi.  and  the  Ed* 
inburgh  Phi!.  Transaction?,  Vol.  n. " 


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